1.2.2: Evaluating, Adding and Subtracting Polynomials
 Page ID
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By the end of this section, you will be able to:
 Determine the degree of polynomials
 Add and subtract polynomials
 Evaluate a polynomial
Before you get started, take this readiness quiz.
1. Subtract \((5n+8)−(2n−1).\)
2. Evaluate \(4(2)(5)2(3)(3)\).
Positive Integer Exponents
Remember that a positive integer exponent indicates repeated multiplication of the same quantity. For example, in the expression \(a^m\), the positive integer exponent
\(m\) tells us how many times we use the base \(a\) as a factor.
\(a^m=\underbrace{a\cdot a\cdot \cdots\cdot a}_{m a'\text{s}}\)
For example,
\((−9)^5=(−9)(−9)(−9)(−9)(−9)\).
Let’s review the vocabulary for expressions with exponents.
\(a^m=\underbrace{a\cdot a\cdots a}_{m a'\text{s}}\)
This is read \(a\) to the \(m\)th (power), or \(a\) to the power \(m\).
In the expression \(a^m\) with positive integer \(m\) and \(a\not=0\), the exponent \(m\) tells us how many times we use the base \(a\) as a factor.
a. Evaluate \(2^3\).
b. Evaluate \(−7^2\).
c. Evaluate \((−1)^4\).
d. Write using exponents: \(−2\cdot 2\cdot 2\).
e. Identify the base and the exponent: \(4^3\).
Solution
a. \(2^3=2\cdot 2\cdot 2=8\)
b. \(−7^2=49\)
c. \((1)^4=(1)\cdot(1)\cdot(1)\cdot(1)=1\)
d. \(2\cdot 2\cdot 2=2^3\)
e. \(4^3\) has an exponent of 3 and the base is 4 since there are no parentheses that indicate including the "".
a. Evaluate \(3^4\).
b. Evaluate \(−2^4\).
c. Evaluate \((2)^3\).
d. Write using exponents \(6\cdot 6\cdot 6\cdot 6\).
e. Identify the base and the exponent: \(2\cdot 5^7\).
 Answer

a. 81
b. 16
c. 8
d. \(6^4\)
e. The exponent is 7 and the base is 5 (since there are no parentheses that would include 2 or 2).
Polynomials
Just like we can add, subtract and multiply numbers, we can also do these things with variables (holding places for numbers). We will next define some words that help us highlight features of resulting expressions, and then, as we did in the case of a linear expression, we will see how to add and subtract.
A monomial is an expression formed by multiplying variables and numbers. A polynomial is a sum of monomials. We could also say that a polynomial is an expression formed by adding (or subtracting) and/or multiplying numbers and variables together.
A term of a polynomial is a monomial that is combined with other monomials using addition or subtraction.
 A polynomial with exactly two terms is called a binomial.
 A polynomial with exactly three terms is called a trinomial.
The coefficient of a term is the number multiplying the product of variables which are then combined via addition.
Here are some examples of polynomials.
Polynomial  \(y+1\)  \(4a^2−7ab+2b^2\)  \(4x^4+x^3+8x^2−9x+1\)  \(0\) 

Monomial  \(14\)  \(8y^2\)  \(−9x^3y^5\)  \(−13a^3b^2c\) 
Binomial  \(a+7ab+7b\)  \(4x^2−y^2\)  \(y^2−16\)  \(3p^3q−9p^2q\) 
Trinomial  \(x^2−7x+12\)  \(9m^2+2mn−8n^2\)  \(6k^4−k^3+8k\)  \(z^4+3z^2−1\) 
Notice that every monomial, binomial, and trinomial is also a polynomial. They are just special members of the “family” of polynomials and so they have special names. We use the words monomial, binomial, and trinomial when referring to these special polynomials and just call all the rest polynomials.
The degree of a polynomial and the degree of its terms are determined by the exponents of the variable. A monomial that has no variable, just a constant, is a special case. The degree of a constant is 0.
 The degree of a term is the sum of the exponents of its variables.
 The degree of a constant is 0.
 The degree of a polynomial is the highest degree among all its terms.
 A polynomial of degree 1 is called a linear expression or polynomial.
 A polynomial of degree 2 is called a quadratic expression or polynomial.
Note that linear expressions from the last chapter are polynomials of degree 1 or 0 (in the case where there is no variable!). Also, linear expressions with one variable are either binomials or monomials. How many terms can a linear expression with two variables have?
Let’s see how this works by looking at several polynomials. We’ll take it step by step, starting with monomials, and then progressing to polynomials with more terms. Let's start by looking at a monomial. The monomial \(8ab^2\) has two variables \(a\) and \(b\). To find the degree we need to find the sum of the exponents. The variable \(a\) doesn't have an exponent written, but remember that means the exponent is 1. The exponent of \(b\) is 2. The sum of the exponents, \(1+2\), is 3, so the degree is 3. The coefficient of the term is 8. Unless it is clear from the context, you should specify the term for which a number is the coefficient in some way. For example, \(8\) is the coefficient of the \(ab^2\)term.
Monomial  \(8ab^2\) 

Coefficient  \(8\) 
Variables  \(a\), \(b\) 
Exponent of \(a\)  \(1\) 
Exponent of \(b\)  \(2\) 
Degree of the monomial  \(1+2=3\) 
Here are some additional examples.
Monomial  \(14\)  \(8ab^2\)  \(9x^3y^5\)  \(13a\) 

Term  \(14\)  \(8ab^2\)  \(9x^3y^5\)  \(13a\) 
Coefficient of term  \(14\)  \(8\)  \(9\)  \(13\) 
Degree of the monomial  \(0\)  \(3\)  \(8\)  \(1\) 
Binomial  \(h+7\)  \(7b^23b\)  \(x^2y^225\)  \(4n^38n^2\) 

Terms  \(h\), \(7\)  \(7b^2\), \(3b\)  \(x^2y^2\), \(25\)  \(4n^3\), \(8n^2\) 
Coefficients or respective terms  \(1\), \(7\)  \(7\), \(3\)  \(1\), \(25\)  \(4\), \(8\) 
Degree of respective terms  \(1, 0\)  \(2, 1\)  \(4, 0\)  \(3, 2\) 
Degree of the binomial  \(1\)  \(2\)  \(4\)  \(3\) 
Trinomial  \(x^212x+27\)  \(9a^2+6ab+b^2\)  \(6m^4m^3n^2+8mn^5\)  \(z^4+3z^21\) 

Terms  \(x^2\), \(12x\), \(27\)  \(9x^2\), \(6ab\), \(b^2\)  \(6m^4\), \(m^3n^2\), \(8mn^5\)  \(z^4\), \(3z^2\), \(1\) 
Coefficient of respective terms  \(1\), \(12\), \(27\) 
\(9\), \(6\), \(1\) 
\(6\), \(1\), \(8\)  \(1\), \(3\), \(1\) 
Degree of respective terms  \(2, 1, 0\)  \(2, 2, 2\)  \(4, 5, 6\)  \(4, 2, 0\) 
Degree of the trinomial  \(2\)  \(2\)  \(6\)  \(4\) 
Polynomial  \(y1\)  \(3y^22y5\)  \(4x^4 +x^3 +8x^29x+1\) 

Terms  \(y\), \(1\)  \(3y^2\), \(2y\), \(5\)  \(4x^4\), \(x^3\), \(8x^2\), \(9x\), \(1\) 
Coefficient of respective terms  \(1\), \(1\)  \(3\), \(2\), \(5\)  \(4\), \(1\), \(8\), \(9\), \(1\) 
Degree of respective terms  \(1, 0\)  \(2, 1, 0\)  \(4, 3, 2, 1, 0\) 
Degree of the polynomial  \(1\)  \(2\)  \(4\) 
Working with polynomials is easier (because its form is consistent) when you list the terms in descending order of degrees. When a polynomial is written this way, it is said to be in standard form of a polynomial. Get in the habit of writing the term with the highest degree first. This can be arranged because you can change the order in which terms are added (carry '' signs along with the terms when rearranging).
For example,
\(\begin{align*} \quad &5x^32xx^5+3\\ =& 5x^3+(2x)+(x^5)+3\\ =& (x^5)+(5x^3)+(2x)+3\\ =& x^55x^32x+3\end{align*}\)
so, the standard form of \(5x^32xx^5+3\) is either \(x^5+(5x^3)+(2x)+3\) or, equivalently, \(x^55x^32x+3.\)
Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial. Then, find the degree of each polynomial.
a. \(7y^2−5y+3\)
b. \(−2a^4b^2\)
c. \(3x^5−4x^3−6x^2+x−8\)
d. \(2y−8xy^3\)
e. \(15\)
Solution
Polynomial  Number of terms  Type  Degree of each term  Degree of the polynomial  

a.  \(7y^2−5y+3\)  3  Trinomial  2, 1, 0  2 
b.  \(−2a^4b^2\)  1  Monomial  6  6 
c.  \(3x^5−4x^3−6x^2+x−8\)  5  Polynomial  5, 3, 2, 1, 0  5 
d.  \(2y−8xy^3\)  2  Binomial  1, 4  4 
e.  \(15\)  1  Monomial  0  0 
Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial. Then, find the degree of each polynomial.
a. \(−5\)
b. \(8y^3−7y^2−y−3\)
c. \(−3x^2y−5xy+9xy^3\)
d. \(81m^2−4n^2\)
e. \(−3x^6y^3z\)
 Answer a

It is a monomial of degree 0.
 Answer b

It is a polynomial of degree 3.
 Answer c

It is a trinomial of degree 4.
 Answer d

It is a binomial of degree 2.
 Answer e

It is a monomial of degree 10.
Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial. Then, find the degree of each polynomial.
a. \(64k^3−8\)
b. \(9m^3+4m^2−2\)
c. \(56\)
d. \(8a^4−7a^3b−6a^2b^2−4ab^3+7b^4\)
e. \(p^4q^3\)
 Answer

a. It is a binomial of degree 3.
b. It is a trinomial of degree 3.
c. It is a monomial of degree 0.
d. It is a polynomial of degree 4.
e. It is a monomial of degree 7.
Evaluating Polynomials and the Order of Operations
When we have an expression that involves exponents, we should evaluate the exponents before multiplying, dividing, adding or subtracting. Also, parentheses should be evaluated before anything else unless you use the distribution property.
So, for example,
\(\begin{align*}\quad &3(2)^32\cdot 3^27\\ =& 3 (8)2(9)7\\ =& 24187\\ =& 49\end{align*}\) 
\(\begin{align*}\quad &27(23)^3\\ =& 27(1)^3\\ =& 27(1)\\ =& 2+7\\ =& 9\end{align*}\) (Note here that you can not distribute the 7 because exponents come before multiplication in the order of operations.) 
\(\begin{align*}\quad &2\left(\dfrac{3}{2}\right)^32\\ =& 2\cdot\dfrac{27}{4\cdot 2}2\\ =& \dfrac{2\cdot 27}{4\cdot 2}2\\ =& \dfrac{27}{4}2\cdot\dfrac{4}{4}\\ =& \dfrac{27}{4}\dfrac{8}{4}\\ =& \dfrac{278}{4}\\ =& \dfrac{19}{4}\quad \text{ or }\quad 4\dfrac{3}{4}.\end{align*}\) 
Note that for the purpose of arithmetic improper fractions are preferable to mixed numbers, but once you want to estimate the value or find the quantity on the number line you may find mixed numbers more convenient. Occasionally you may find addition and subtraction can be simplified by a mixed approach.
Just as in the case of linear expressions (polynomials of degree \(1\)) we can evaluate polynomials by substituting in values of the variables. For example,
Polynomial  Values of the variables  Evaluating the polynomial 

\(3x^32x^24\)  \(x=2\) 
\(\begin{align*}\quad &3x^32x^24\\ =& 3(2)^32(2)^24\\ =& 3(8)2(4)4\\ =& 2484\\ =& 36\end{align*}\) 
\(a^2b+ab4\)  \(a=2\), \(b=1\) 
\(\begin{align*}\quad &a^2b+ab4\\ =& (2)^2(1)+(2)(1)4\\ =& 4(1)24\\ =& 424\\ =& 2\end{align*}\) 
\(b^24ac\)  \(a=2, b=2\), \(c=1\) 
\(\begin{align*}\quad &b^24ac\\ =& (2)^24(2)(1)\\ =& 48\\ =& 4\end{align*}\) 
Evaluate the trinomial \(5x^2−8x+4\) at:
a. \(x=4\)
b. \(x=2\)
c. \(x=0\)
Solution
a.
\(\quad 5x^2−8x+4\)  

Substitute 4 for \(x\). 
\(\quad 5(4)^2−8(4)+4\) 
Simplify the exponents.  \(=5(16)−8(4)+4\) 
Multiply. 
\(=8032+4\) 
Simplify.  \(=52\) 
b.
\(\quad 5x^2−8x+4\)  

Substitute \(2\) for \(x\).  \(\quad 5(2)^2−8(2)+4\) 
Simplify the exponents.  \(= 5(4)−8(2)+4\) 
Multiply. 
\(= 20+16+4\) 
Simplify.  \(=40\) 
c.
\(\quad 5x^2−8x+4\)  

Substitute \(0\) for \(x\).  \(\quad 5(0)^2−8(0)+4\) 
Simplify the exponents.  \(=5(0)−8(0)+4\) 
Multiply.  \(=0+0+4\) 
Simplify.  \(=4\) 
Evaluate the trinomial \(3x^2+2x−15\) at:
a. \(x=3\)
b. \(x=5\)
c. \(x=0\)
 Answer

a. \(18\)
b. \(50\)
c. \(−15\)
Evaluate the trinomial \(5x^2−x−4\) at:
a. \(x=2\)
b. \(x=1\)
c. \(x=0\)
 Answer

a. \(18\)
b. \(2\)
c. \(−4\)
Adding and Subtracting Polynomials
Just like we could add and subtract linear expressions (polynomials of degree 1) we can add and subtract polynomials in general by combining like terms.
Two monomials are 'like' if one monomial is a constant times the other. For example,
\(3x^2y^5\) and \(2y^5x^2\) are like because \(x^2y^5\) is equivalent to \(y^5x^2\) since multiplication of numbers and therefore variables can be done in any order (for example, \(2^23^5=3^52^2\)) and so \(3 x^2 y^5=\dfrac{3}{2}(2y^5x^2)\).
You can also think of like terms as terms that have the same variables to the same exponents so that the 'variable parts' are equivalent. Or, two terms are like if they are equivalent aside from their coefficients (numbers multiplying the variables).
We can add and subtract like terms. For example,
\(3x^2y^5+(2y^5x^2)=x^2y^5\qquad\) (3 dimes  2 dimes is 1 dime)
and
\(3x^2y^5(2y^5x^2)=5x^2y^5\qquad\) (3 dimes + 2 dimes is 5 dimes).
If the monomials are like terms, we just combine them by adding or subtracting their coefficients.
Add or subtract:
a. \(25y^2+15y^2\)
b. \(16pq^3−(−7pq^3)\)
Solution
a.
\(\quad 25y^2+15y^2\)  

Identify like terms.  The like terms are \(25y^2\) and \(15y^2\). 
Combine like terms.  \(=40y^2\) 
b.
\(\quad 16pq^3−(−7pq^3)\)  

Rewrite without the parentheses.  \(=16pq^3+7pq^3\) 
Identify like terms.  The like terms are \(16pq^3\) and \(7pq^3\). 
Combine like terms.  \(=23pq^3\) 
Add or subtract:
a. \(12q^2+9q^2\)
b. \(8mn^3−(−5mn^3)\)
 Answer

a. \(21q^2\)
b. \(13mn^3\)
Add or subtract:
a. \(−15c^2+8c^2\)
b. \(−15y^2z^3−(−5y^2z^3)\)
 Answer

a. \(−7c^2\)
b. \(−10y^2z^3\)
Remember that like terms must have the same variables with the same exponents.
Simplify:
a. \(a^2+7b^2−6a^2\)
b. \(u^2v+5u^2−3v^2\)
Solution
a.
\(\quad a^2+7b^2−6a^2\)  

Identify like terms.  The like terms are \(a^2\) and \(6a^2\). 
Combine like terms.  \(= −5a^2+7b^2\) 
b.
\(u^2v+5u^2−3v^2\)  

Identify like terms.  none 
Combine like terms.  There are no like terms to combine. In this case, the polynomial is unchanged. \(u^2v+5u^2−3v^2\) 
Add:
a. \(8y^2+3z^2−3y^2\)
b. \(m^2n^2−8m^2+4n^2\)
 Answer

a. \(5y^2+3z^2\)
b. \(m^2n^2−8m^2+4n^2\)
Add:
a. \(3m^2+n^2−7m^2\)
b. \(pq^2−6p−5q^2\)
 Answer

a. \(−4m^2+n^2\)
b. \(pq^2−6p−5q^2\)
We can think of adding and subtracting polynomials as just adding and subtracting a series of monomials. Look for the like terms—those with the same variables and the same exponent. The Commutative Property allows us to rearrange the terms to put like terms together.
Find the sum \((7y^2−2y+9) + (4y^2−8y−7)\).
Solution
\(\quad (7y^2−2y+9)+(4y^2−8y−7)\)  

Rewrite without parentheses.  \(=7y^2−2y+9+4y^2−8y−7\) 
Identify like terms. 
The like terms are

Rearrange to get the like terms together.  \(=7y^2+4y^22y−8y+9−7\) 
Combine like terms.  \(=11y^2−10y+2 \) 
Find the sum \( (7x^2−4x+5)+(x^2−7x+3)\).
 Answer

\(8x^2−11x+8\)
Find the sum \((14y^2+6y−4)+(3y^2+8y+5)\).
 Answer

\(17y^2+14y+1\)
Be careful with the signs as you distribute while subtracting the polynomials in the next example. As with linear expressions, we subtract a polynomial by adding its opposite which is obtained by negating each term.
Find the difference \((9w^2−7w+5)−(2w^2−4)\).
Solution
\(\quad (9w^2−7w+5)−(2w^2−4)\)  

Rewrite without parentheses.  \(=9w^2−7w+5−2w^2+4\) 
Distribute and identify like terms. 
The like terms are:

Rearrange to get the like terms together.  \(= 9w^22w^2−7w+5+4\) 
Combine like terms.  \(= 7w^2−7w+9 \) 
Find the difference \((8x^2+3x−19) − (7x^2−14)\).
 Answer

\(x^2+3x−5\)
Find the difference \((9b^2−5b−4) − (3b^2−5b−7)\).
 Answer

\(6b^2+3\)
Subtract \(p^2+10pq−2q^2\) from \(p^2+q^2\).
Solution
\(\quad (p^2+q^2) − (p^2+10pq−2q^2)\)  

Rewrite without parentheses.  \(=p^2+q^2p^210pq+2q^2\) 
Identify like terms. 
The like terms are:

Rearrange to get like terms together.  \(= p^2p^2 +q^2 + 2q^2−10pq\) 
Combine like terms.  \(=−10pq+3q^2\) 
Subtract \(a^2+5ab−6b^2\) from \(a^2+b^2\).
 Answer

\(−5ab+7b^2\)
Subtract \(m^2−7mn−3n^2\) from \(m^2+n^2\).
 Answer

\(7mn+4n^2\)
Find the sum \((u^2−6uv+5v^2) + (3u^2+2uv)\).
Solution
\(\quad (u^2−6uv+5v^2) + (3u^2+2uv)\)  

Distribute and identify like terms.  \(=\underline{\underline{u^2}} \underline{6uv}+5v^2+\underline{\underline{3u^2}}+ \underline{2uv}\) 
Rearrange the terms to put like terms together.  \(=\underline{\underline{u^2+3u^2}} \underline{6uv+ 2uv}+5v^2\) 
Combine like terms.  \(=4u^2−4uv+5v^2\) 
Find the sum \((3x^2−4xy+5y^2) + (2x^2−xy)\).
 Answer

\(5x^2−5xy+5y^2\)
Find the sum \((2x^2−3xy−2y^2) + (5x^2−3xy)\).
 Answer

\(7x^2−6xy−2y^2\)
When we add and subtract more than two polynomials, the process is the same.
Simplify \((a^3−a^2b) − (ab^2+b^3) + (a^2b+ab^2)\).
Solution
\(\quad (a^3−a^2b) − (ab^2+b^3) + (a^2b+ab^2)\)  

Distribute.  \(=a^3 a^2bab^2 b^3 +a^2b+ab^2\) 
Identify like terms. 
The like terms are:

Rearrange the terms to put like terms together.  \(=a^3−a^2b + a^2b− ab^2 + ab^2  b^3\) 
Combine like terms.  \(=a^3−b^3\) 
Simplify \((x^3−x^2y) − (xy^2+y^3) + (x^2y+xy^2)\).
 Answer

\(x^3+y^3\)
Simplify \((p^3−p^2q) + (pq^2+q^3) − (p^2q+pq^2)\).
 Answer

\(p^3−2p^2q+q^3\)
Applications
Polynomials similar to the one in the next example are used in many fields to model the height of an object at some time after it is projected into the air. The polynomial in the next example is used specifically to model the height of an object which is dropped from 250 ft.
The polynomial \(−16t^2+250\) gives the height of a ball \(t\) seconds after it is dropped from a 250foot tall building. Find the height after \(t=2\) seconds. In this example the variable \(t\) is a place holder for any number you may be interested in rather than an 'unknown' number.
Solution
The interpretation of the expression when \(t=2\) is when evaluated it gives the height of the ball \(2\) seconds after it is dropped from a 250foot tall building.
\(\quad −16t^2+250\)  

To find the height at \(2\) seconds, we substitute \(2\) for \(t\).  \(\quad −16(2)^2+250\) 
Simplify.  \(=−16(4)+250\) 
Simplify.  \( =−64+250\) 
Simplify.  \( =186\) 
Answer the question.  After 2 seconds, the height of the ball is 186 feet. 
The polynomial \(−16t^2+150\) gives the height of a stone \(t\) seconds after it is dropped from a 150foot tall cliff. Find the height after \(t=0\) seconds (the initial height of the object).
 Answer

The height is \(150\) feet.
The polynomial \(−16t^2+175\) gives the height of a ball \(t\) seconds after it is dropped from a 175foot tall bridge. Find the height after \(t=3\) seconds.
 Answer

The height is \(31\) feet.
 Why are \(3x^2y^3\) and \(2y^3x^2\) like terms and why can you add them to get \(x^2y^3\)?
 Why is \(x^2+x\) the opposite of \(x^2x\) ?
 What degree might the sum of a third degree and a 4th degree polynomial be?
 What degree might the sum of two third degree polynomials be?
Simplify \((2x^2+3x1)(3x^25x+2)\).
Key Concepts
 Monomial
 A monomial is an algebraic expression with one term.
 A monomial in one variable is a term of the form \(ax^m\) where \(a\) is a constant and \(m\) is a whole number.
 Polynomials
 Polynomial—A monomial, or two or more algebraic terms combined by addition or subtraction is a polynomial.
 Monomial —A polynomial with exactly one term is called a monomial.
 Binomial — A polynomial with exactly two terms is called a binomial.
 Trinomial —A polynomial with exactly three terms is called a trinomial.
 Degree of a Polynomial
 The degree of a term is the sum of the exponents of its variables.
 The degree of a constant is 0.
 The degree of a polynomial is the highest degree of all its terms.
 A linear expression is a polynomial of degree 0 or 1.
 Term
 Coefficient
 Evaluating, Adding/Subtracting polynomials