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Mathematics LibreTexts

11.6: Binomial Theorem

Binomial Theorem
In this section, you will:
  • Apply the Binomial Theorem.

A polynomial with two terms is called a binomial. We have already learned to multiply binomials and to raise binomials to powers, but raising a binomial to a high power can be tedious and time-consuming. In this section, we will discuss a shortcut that will allow us to find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> (x+y) n  without multiplying the binomial by itself <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>n</mi></annotation-xml></semantics></math> times.

Identifying Binomial Coefficients

In Counting Principles, we studied combinations. In the shortcut to finding<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> (x+y) n , we will need to use combinations to find the coefficients that will appear in the expansion of the binomial. In this case, we use the notation<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="true">(</mo><mtable/></mrow></annotation-xml></semantics></math> n r )  instead of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo><mo>,</mo></mrow></annotation-xml></semantics></math> but it can be calculated in the same way. So

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> n r )=C(n,r)= n! r!(n−r)!  

 

The combination<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="true">(</mo><mtable/></mrow></annotation-xml></semantics></math> n r ) is called a binomial coefficient. An example of a binomial coefficient is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="true">(</mo><mtable/></mrow></annotation-xml></semantics></math> 5 2 )=C(5,2)=10. 

Binomial Coefficients

If <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi></mrow></annotation-xml></semantics></math> and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi></mrow></annotation-xml></semantics></math>are integers greater than or equal to 0 with <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi><mo>≥</mo><mi>r</mi><mo>,</mo></mrow></annotation-xml></semantics></math> then the binomial coefficient is

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> n r )=C(n,r)= n! r!(n−r)!

Is a binomial coefficient always a whole number?

Yes. Just as the number of combinations must always be a whole number, a binomial coefficient will always be a whole number.

Finding Binomial Coefficients

Find each binomial coefficient.

  1. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 5 3 )
  2. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 9 2 )
  3. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 9 7 )

Use the formula to calculate each binomial coefficient. You can also use the <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>n</mi></msub></mrow></annotation-xml></semantics></math> C r function on your calculator.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> n r )=C(n,r)= n! r!(n−r)!
  1. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 5 3 )= 5! 3!(5−3)! = 5⋅4⋅3! 3!2! =10
  2. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 9 2 )= 9! 2!(9−2)! = 9⋅8⋅7! 2!7! =36
  3. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 9 7 )= 9! 7!(9−7)! = 9⋅8⋅7! 7!2! =36
Analysis

Notice that we obtained the same result for parts (b) and (c). If you look closely at the solution for these two parts, you will see that you end up with the same two factorials in the denominator, but the order is reversed, just as with combinations.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> n r )=( n n−r )

 

Find each binomial coefficient.

 

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 7 3 ) 
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 11 4 ) 

 

 

35
330

 

Using the Binomial Theorem

When we expand <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></msup></mrow></annotation-xml></semantics></math> n by multiplying, the result is called a binomial expansion, and it includes binomial coefficients. If we wanted to expand <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></msup></mrow></annotation-xml></semantics></math> 52 , we might multiply <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math> by itself fifty-two times. This could take hours! If we examine some simple binomial expansions, we can find patterns that will lead us to a shortcut for finding more complicated binomial expansions.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></msup></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 2 = x 2 +2xy+ y 2 (x+y) 3 = x 3 +3 x 2 y+3x y 2 + y 3 (x+y) 4 = x 4 +4 x 3 y+6 x 2 y 2 +4x y 3 + y 4

First, let’s examine the exponents. With each successive term, the exponent for <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>x</mi></mrow></annotation-xml></semantics></math> decreases and the exponent for <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>y</mi></mrow></annotation-xml></semantics></math> increases. The sum of the two exponents is <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi></mrow></annotation-xml></semantics></math> for each term.

Next, let’s examine the coefficients. Notice that the coefficients increase and then decrease in a symmetrical pattern. The coefficients follow a pattern:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> n 0 ),( n 1 ),( n 2 ),...,( n n ).

These patterns lead us to the Binomial Theorem, which can be used to expand any binomial.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></msup></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> n = ∑ k=0 n ( n k ) x n−k y k = x n +( n 1 ) x n−1 y+( n 2 ) x n−2 y 2 +...+( n n−1 )x y n−1 + y n

Another way to see the coefficients is to examine the expansion of a binomial in general form,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>+</mo><mi>y</mi><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>to successive powers 1, 2, 3, and 4.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></msup></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 1 =x+y (x+y) 2 = x 2 +2xy+ y 2 (x+y) 3 = x 3 +3 x 2 y+3x y 2 + y 3 (x+y) 4 = x 4 +4 x 3 y+6 x 2 y 2 +4x y 3 + y 4

Can you guess the next expansion for the binomial<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> (x+y) 5 ? 

<figure id="CNX_Precalc_Figure_11_06_002">Graph of the function f_2.</figure>

See [link], which illustrates the following:

  • There are <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></annotation-xml></semantics></math> terms in the expansion of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></msup></mrow></annotation-xml></semantics></math> n .
  • The degree (or sum of the exponents) for each term is <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi><mo>.</mo></mrow></annotation-xml></semantics></math>
  • The powers on <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>x</mi></mrow></annotation-xml></semantics></math> begin with <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi></mrow></annotation-xml></semantics></math> and decrease to 0.
  • The powers on <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>y</mi></mrow></annotation-xml></semantics></math> begin with 0 and increase to <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi><mo>.</mo></mrow></annotation-xml></semantics></math>
  • The coefficients are symmetric.

To determine the expansion on <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></msup></mrow></annotation-xml></semantics></math> 5 , we see <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi><mo>=</mo><mn>5</mn><mo>,</mo></mrow></annotation-xml></semantics></math> thus, there will be 5+1 = 6 terms. Each term has a combined degree of 5. In descending order for powers of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>x</mi><mo>,</mo></mrow></annotation-xml></semantics></math> the pattern is as follows:

  • Introduce <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>x</mi></msup></mrow></annotation-xml></semantics></math> 5 , and then for each successive term reduce the exponent on <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>x</mi></mrow></annotation-xml></semantics></math> by 1 until <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>x</mi></msup></mrow></annotation-xml></semantics></math> 0 =1 is reached.
  • Introduce <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>y</mi></msup></mrow></annotation-xml></semantics></math> 0 =1, and then increase the exponent on <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>y</mi></mrow></annotation-xml></semantics></math> by 1 until <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>y</mi></msup></mrow></annotation-xml></semantics></math> 5 is reached.
    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>x</mi></msup></mrow></annotation-xml></semantics></math> 5 ,   x 4 y,   x 3 y 2 ,   x 2 y 3 ,  x y 4 ,   y 5

The next expansion would be

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></msup></mrow></annotation-xml></semantics></math> 5 = x 5 +5 x 4 y+10 x 3 y 2 +10 x 2 y 3 +5x y 4 + y 5 .

But where do those coefficients come from? The binomial coefficients are symmetric. We can see these coefficients in an array known as Pascal's Triangle, shown in [link].

<figure class="medium" id="CNX_Precalc_Figure_11_06_001">Pascal's Triangle</figure>

To generate Pascal’s Triangle, we start by writing a 1. In the row below, row 2, we write two 1’s. In the 3rd row, flank the ends of the rows with 1’s, and add <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>1</mn><mo>+</mo><mn>1</mn></mrow></annotation-xml></semantics></math> to find the middle number, 2. In the <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi><mtext>th</mtext></mrow></annotation-xml></semantics></math> row, flank the ends of the row with 1’s. Each element in the triangle is the sum of the two elements immediately above it.

To see the connection between Pascal’s Triangle and binomial coefficients, let us revisit the expansion of the binomials in general form.

Pascal's Triangle expanded to show the values of the triangle as x and y terms with exponents
The Binomial Theorem

The Binomial Theorem is a formula that can be used to expand any binomial.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></msup></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> n = ∑ k=0 n ( n k ) x n−k y k = x n +( n 1 ) x n−1 y+( n 2 ) x n−2 y 2 +...+( n n−1 )x y n−1 + y n

Given a binomial, write it in expanded form.

  1. Determine the value of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi></mrow></annotation-xml></semantics></math>according to the exponent.
  2. Evaluate the <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow></annotation-xml></semantics></math> through <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>k</mi><mo>=</mo><mi>n</mi></mrow></annotation-xml></semantics></math> using the Binomial Theorem formula.
  3. Simplify.
Expanding a Binomial

Write in expanded form.

  1. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> (x+y) 5  
  2. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> ( 3x−y ) 4  
  1. Substitute <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi><mo>=</mo><mn>5</mn></mrow></annotation-xml></semantics></math> into the formula. Evaluate the <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow></annotation-xml></semantics></math> through <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>k</mi><mo>=</mo><mn>5</mn></mrow></annotation-xml></semantics></math> terms. Simplify.
    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></msup></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 5 =( 5 0 ) x 5 y 0 +( 5 1 ) x 4 y 1 +( 5 2 ) x 3 y 2 +( 5 3 ) x 2 y 3 +( 5 4 ) x 1 y 4 +( 5 5 ) x 0 y 5 (x+y) 5 = x 5 +5x 4 y+10 x 3 y 2 +10 x 2 y 3 +5x y 4 + y 5
  2. Substitute <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi><mo>=</mo><mn>4</mn></mrow></annotation-xml></semantics></math> into the formula. Evaluate the <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow></annotation-xml></semantics></math> through <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>k</mi><mo>=</mo><mn>4</mn></mrow></annotation-xml></semantics></math> terms. Notice that <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>3</mn><mi>x</mi></mrow></annotation-xml></semantics></math> is in the place that was occupied by <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>x</mi></mrow></annotation-xml></semantics></math> and that <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>–</mo><mi>y</mi></mrow></annotation-xml></semantics></math> is in the place that was occupied by <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>y</mi><mo>.</mo></mrow></annotation-xml></semantics></math> So we substitute them. Simplify.
    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>3</mn><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></msup></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 4 =( 4 0 ) (3x) 4 (−y) 0 +( 4 1 ) (3x) 3 (−y) 1 +( 4 2 ) (3x) 2 (−y) 2 +( 4 3 ) (3x) 1 (−y) 3 +( 4 4 ) (3x) 0 (−y) 4(3x−y) 4 =81 x 4 −108 x 3 y+54 x 2 y 2 −12x y 3 + y 4
Analysis

Notice the alternating signs in part b. This happens because<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="false">(</mo><mo>−</mo><mi>y</mi><mo stretchy="false">)</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>raised to odd powers is negative, but<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="false">(</mo><mo>−</mo><mi>y</mi><mo stretchy="false">)</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>raised to even powers is positive. This will occur whenever the binomial contains a subtraction sign.

Write in expanded form.

  1. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></msup></mrow></annotation-xml></semantics></math> 5
  2. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi><mo stretchy="false">)</mo></mrow></msup></mrow></annotation-xml></semantics></math> 3
  1. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>x</mi></msup></mrow></annotation-xml></semantics></math> 5 −5 x 4 y+10 x 3 y 2 −10 x 2 y 3 +5x y 4 − y 5
  2. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>8</mn><msup/></mrow></annotation-xml></semantics></math> x 3 +60 x 2 y+150x y 2 +125 y 3

Using the Binomial Theorem to Find a Single Term

Expanding a binomial with a high exponent such as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> (x+2y) 16  can be a lengthy process.

Sometimes we are interested only in a certain term of a binomial expansion. We do not need to fully expand a binomial to find a single specific term.

Note the pattern of coefficients in the expansion of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> (x+y) 5 .

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></msup></mrow></annotation-xml></semantics></math> 5 = x 5 +( 5 1 ) x 4 y+( 5 2 ) x 3 y 2 +( 5 3 ) x 2 y 3 +( 5 4 )x y 4 + y 5

The second term is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 5 1 ) x 4 y. The third term is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 5 2 ) x 3 y 2 . We can generalize this result.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> n r ) x n−r y r
The (r+1)th Term of a Binomial Expansion

The<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="false">(</mo><mi>r</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mtext>th</mtext><mtext> </mtext></mrow></annotation-xml></semantics></math>term of the binomial expansion of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> (x+y) n  is:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> n r ) x n−r y r

Given a binomial, write a specific term without fully expanding.

  1. Determine the value of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi></mrow></annotation-xml></semantics></math> according to the exponent.
  2. Determine <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo stretchy="false">(</mo><mi>r</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>.</mo></mrow></annotation-xml></semantics></math>
  3. Determine <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>.</mo></mrow></annotation-xml></semantics></math>
  4. Replace <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi></mrow></annotation-xml></semantics></math> in the formula for the <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo stretchy="false">(</mo><mi>r</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mtext>th</mtext></mrow></annotation-xml></semantics></math> term of the binomial expansion.
Writing a Given Term of a Binomial Expansion

Find the tenth term of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> (x+2y) 16  without fully expanding the binomial.

Because we are looking for the tenth term, <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>10</mn><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math> we will use <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>=</mo><mn>9</mn></mrow></annotation-xml></semantics></math> in our calculations.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> n r ) x n−r y r
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 16 9 ) x 16−9 (2y) 9 =5,857,280 x 7 y 9

Find the sixth term of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> (3x−y) 9  without fully expanding the binomial.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo>−</mo><mn>10</mn><mo>,</mo><mn>206</mn><msup/></mrow></annotation-xml></semantics></math> x 4 y 5

Access these online resources for additional instruction and practice with binomial expansion.

Key Equations

Binomial Theorem <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></msup></mrow></annotation-xml></semantics></math> n = ∑ k−0 n ( n k ) x n−k y k
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo stretchy="false">(</mo><mi>r</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mi>t</mi><mi>h</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>term of a binomial expansion <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> n r ) x n−r y r

Key Concepts

  • <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> n r ) is called a binomial coefficient and is equal to <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>See [link].
  • The Binomial Theorem allows us to expand binomials without multiplying. See [link].
  • We can find a given term of a binomial expansion without fully expanding the binomial. See [link].

Section Exercises

Verbal

What is a binomial coefficient, and how it is calculated?

A binomial coefficient is an alternative way of denoting the combination <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>C</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo stretchy="false">).</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>It is defined as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> n r )= C(n,r) = n! r!(n−r)! .

What role do binomial coefficients play in a binomial expansion? Are they restricted to any type of number?

What is the Binomial Theorem and what is its use?

The Binomial Theorem is defined as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> (x+y) n = ∑ k=0 n ( n k ) x n−k y k  and can be used to expand any binomial.

When is it an advantage to use the Binomial Theorem? Explain.

Algebraic

For the following exercises, evaluate the binomial coefficient.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 6 2 )

15

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 5 3 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 7 4 )

35

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 9 7 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 10 9 )

10

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 25 11 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 17 6 )

12,376

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 200 199 )

For the following exercises, use the Binomial Theorem to expand each binomial.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>4</mn><mi>a</mi><mo>−</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msup></mrow></annotation-xml></semantics></math> 3

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>64</mn><msup/></mrow></annotation-xml></semantics></math> a 3 −48 a 2 b+12a b 2 − b 3

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>5</mn><mi>a</mi><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msup></mrow></annotation-xml></semantics></math> 3

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>3</mn><mi>a</mi><mo>+</mo><mn>2</mn><mi>b</mi><mo stretchy="false">)</mo></mrow></msup></mrow></annotation-xml></semantics></math> 3

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>27</mn><msup/></mrow></annotation-xml></semantics></math> a 3 +54 a 2 b+36a b 2 +8 b 3

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo stretchy="false">)</mo></mrow></msup></mrow></annotation-xml></semantics></math> 4

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo stretchy="false">)</mo></mrow></msup></mrow></annotation-xml></semantics></math> 5

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>1024</mn><msup/></mrow></annotation-xml></semantics></math> x 5 +2560 x 4 y+2560 x 3 y 2 +1280 x 2 y 3 +320x y 4 +32 y 5

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>3</mn><mi>x</mi><mo>−</mo><mn>2</mn><mi>y</mi><mo stretchy="false">)</mo></mrow></msup></mrow></annotation-xml></semantics></math> 4

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>3</mn><mi>y</mi><mo stretchy="false">)</mo></mrow></msup></mrow></annotation-xml></semantics></math> 5

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>1024</mn><msup/></mrow></annotation-xml></semantics></math> x 5 −3840 x 4 y+5760 x 3 y 2 −4320 x 2 y 3 +1620x y 4 −243 y 5

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></annotation-xml></semantics></math> 1 x +3y ) 5

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mo stretchy="false">(</mo><msup/></mrow></msup></mrow></annotation-xml></semantics></math> x −1 +2 y −1 ) 4

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>1</mn></mfrac></mrow></annotation-xml></semantics></math> x 4 + 8 x 3 y + 24 x 2 y 2 + 32 x y 3 + 16 y 4

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mo stretchy="false">(</mo><msqrt/></mrow></msup></mrow></annotation-xml></semantics></math> x − y ) 5

For the following exercises, use the Binomial Theorem to write the first three terms of each binomial.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msup></mrow></annotation-xml></semantics></math> 17

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>a</mi></msup></mrow></annotation-xml></semantics></math> 17 +17 a 16 b+136 a 15 b 2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup></mrow></annotation-xml></semantics></math> 18

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>−</mo><mn>2</mn><mi>b</mi><mo stretchy="false">)</mo></mrow></msup></mrow></annotation-xml></semantics></math> 15

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>a</mi></msup></mrow></annotation-xml></semantics></math> 15 −30 a 14 b+420 a 13 b 2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>2</mn><mi>y</mi><mo stretchy="false">)</mo></mrow></msup></mrow></annotation-xml></semantics></math> 8

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>3</mn><mi>a</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msup></mrow></annotation-xml></semantics></math> 20

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>3</mn><mo>,</mo><mn>486</mn><mo>,</mo><mn>784</mn><mo>,</mo><mn>401</mn><msup/></mrow></annotation-xml></semantics></math> a 20 +23,245,229,340 a 19 b+73,609,892,910 a 18 b 2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>a</mi><mo>+</mo><mn>4</mn><mi>b</mi><mo stretchy="false">)</mo></mrow></msup></mrow></annotation-xml></semantics></math> 7

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mo stretchy="false">(</mo><msup/></mrow></msup></mrow></annotation-xml></semantics></math> x 3 − y ) 8

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>x</mi></msup></mrow></annotation-xml></semantics></math> 24 −8 x 21 y +28 x 18 y

For the following exercises, find the indicated term of each binomial without fully expanding the binomial.

The fourth term of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> (2x−3y) 4

The fourth term of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> (3x−2y) 5

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>720</mn><msup/></mrow></annotation-xml></semantics></math> x 2 y 3

The third term of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> (6x−3y) 7

The eighth term of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> (7+5y) 14

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>220</mn><mo>,</mo><mn>812</mn><mo>,</mo><mn>466</mn><mo>,</mo><mn>875</mn><mo>,</mo><mn>000</mn><msup/></mrow></annotation-xml></semantics></math> y 7

The seventh term of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> (a+b) 11

The fifth term of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> (x−y) 7

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>35</mn><msup/></mrow></annotation-xml></semantics></math> x 3 y 4

The tenth term of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> (x−1) 12

The ninth term of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> (a−3 b 2 ) 11

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>1</mn><mo>,</mo><mn>082</mn><mo>,</mo><mn>565</mn><msup/></mrow></annotation-xml></semantics></math> a 3 b 16

The fourth term of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> ( x 3 − 1 2 ) 10

The eighth term of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> ( y 2 + 2 x ) 9

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mn>1152</mn><msup/></mrow></mfrac></mrow></annotation-xml></semantics></math> y 2 x 7

Graphical

For the following exercises, use the Binomial Theorem to expand the binomial <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> (x+3) 4 . Then find and graph each indicated sum on one set of axes.

Find and graph<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> f 1 (x), such that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> f 1 (x) is the first term of the expansion.

Find and graph<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> f 2 (x), such that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> f 2 (x) is the sum of the first two terms of the expansion.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>f</mi></msub></mrow></annotation-xml></semantics></math> 2 (x)= x 4 +12 x 3

Graph of the function f_2.

Find and graph<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> f 3 (x), such that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> f 3 (x)  is the sum of the first three terms of the expansion.

Find and graph<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> f 4 (x), such that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> f 4 (x) is the sum of the first four terms of the expansion.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>f</mi></msub></mrow></annotation-xml></semantics></math> 4 (x)= x 4 +12 x 3 +54 x 2 +108x

Graph of the function f_4.

Find and graph<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> f 5 (x), such that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> f 5 (x) is the sum of the first five terms of the expansion.

Extensions

In the expansion of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> (5x+3y) n , each term has the form<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> n k ) a n–k b k , where k successively takes on the value<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mtext> </mtext><mn>...</mn><mo>,</mo><mtext> </mtext><mi>n</mi><mo>.</mo></mrow></annotation-xml></semantics></math>If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> n k )=( 7 2 ), what is the corresponding term?

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>590</mn><mo>,</mo><mn>625</mn><msup/></mrow></annotation-xml></semantics></math> x 5 y 2

In the expansion of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> ( a+b ) n , the coefficient of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> a n−k b k  is the same as the coefficient of which other term?

Consider the expansion of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> (x+b) 40 . What is the exponent of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>b</mi></mrow></annotation-xml></semantics></math> in the <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>k</mi><mtext>th</mtext></mrow></annotation-xml></semantics></math> term?

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></annotation-xml></semantics></math>

Find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> n k−1 )+( n k ) and write the answer as a binomial coefficient in the form<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> n k ). Prove it. Hint: Use the fact that, for any integer<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>p</mi><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>such that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>p</mi><mo>≥</mo><mn>1</mn><mo>,</mo><mtext> </mtext><mi>p</mi><mo>!</mo><mo>=</mo><mi>p</mi><mo stretchy="false">(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>!</mo><mtext>.</mtext></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> n k−1 )+( n k )=( n+1 k ); Proof:

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtable columnalign="left"><mtr/><mtr/><mtr/><mtr><mtd><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mrow><mo>(</mo></mrow></mtd></mtr></mtable></annotation-xml></semantics></math> n k−1 )+( n k ) = n! k!( n−k )! + n! ( k−1 )!( n−( k−1 ) )! = n! k!(n−k)! + n! ( k−1 )!( n−k+1 )! = ( n−k+1 )n! ( n−k+1 )k!(n−k)! + kn! k( k−1 )!( n−k+1 )! = ( n−k+1 )n!+kn! k!( n−k+1 )! = ( n+1 )n! k!( ( n+1 )−k )! = ( n+1 )! k!( ( n+1 )−k )! =( n+1k )

Which expression cannot be expanded using the Binomial Theorem? Explain.

  • <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo stretchy="false">(</mo><msup/></mrow></annotation-xml></semantics></math> x 2 −2x+1)
  • <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mo stretchy="false">(</mo><msqrt/></mrow></msup></mrow></annotation-xml></semantics></math> a +4 a −5) 8
  • <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mo stretchy="false">(</mo><msup/></mrow></msup></mrow></annotation-xml></semantics></math> x 3 +2 y 2 −z) 5
  • <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>3</mn><msup/></mrow></msup></mrow></annotation-xml></semantics></math> x 2 − 2 y 3 ) 12

The expression<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> ( x 3 +2 y 2 −z) 5  cannot be expanded using the Binomial Theorem because it cannot be rewritten as a binomial.

Glossary

binomial coefficient
the number of ways to choose r objects from n objects where order does not matter; equivalent to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>C</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>denoted<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> n r )
binomial expansion
the result of expanding<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> (x+y) n  by multiplying
Binomial Theorem
a formula that can be used to expand any binomial