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7.3: Simplifying Rational Expressions

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Any time you divide a polynomial by a second polynomial, you form what is You will find that material quite helpful for this section. known as a rational expression.

Note

Readers are strongly encouraged to review the material on fractions presented in Section 3 of Chapter 1.

Rational expression

The expression p(x)q(x) where p(x) and q(x) are polynomials, is called a rational expression.

For example, each of the following is a rational expression.

  1. x+23x
  2. x+3x22x4
  3. 2x3y2

In example a), the rational expression is composed of a binomial over a monomial. Example b) is constructed by dividing a binomial by a trinomial. Example c) is composed of a monomial over a monomial, the type of rational expression that will gain the most attention in this section.

Multiplying and Dividing Rational Expressions

We will concentrate on rational expressions with monomial numerators and denominators. Recall that to form the product of two rational numbers, we simply multiply numerators and denominators. The same technique is used to multiply any two rational expressions.

Multiplying rational expressions

Given a/b and c/d, their product is defined as:abcd=acbd

Remember, you need only multiply numerators and denominators. For example:

  1. x32y=2x3y
  2. 2a3b25a9b3=10a227b5
  3. x2y(3x4y2)=3x28y3

Of course, as the next example shows, sometimes you also need to reduce your answer to lowest terms.

Example 7.3.1

Simplify: 2xx24.

Solution

Multiply numerators and denominators.

2xx24=2x54x3

Now, there several different ways you can reduce this answer to lowest terms, two of which are shown below.

You can factor numerator and denominator, then cancel common factors. 2x54x3=2xxxxx22xxx=2xxxxx22xxx=x22

Or you can write the answer as a product, repeat the base and subtract exponents.2x54x3=24x5x3=12x53=12x2

As dividing by 2 is the same as multiplying by 1/2, these answers are equivalent. Also, note that the right-hand method is more efficient

Exercise 7.3.1

Simply: 9x2x6.

Answer

32x

Recall that when dividing fractions, we invert the second fraction and multiply.

Dividing rational expressions

Given a/b and c/d, their quotient is defined as:ab÷cd=abdc=adbc

Example 7.3.2

Simplify: x2y÷x42y2.

Solution

Invert, then multiply.

x2y÷x42y2=x2y2y2x4=2x2y2x4y

Now, there several different ways you can reduce this answer to lowest terms, two of which are shown below.

You can factor numerator and denominator, then cancel common factors. 2x2y2x4y=2xxyyxxxxy=2xxyyxxxxy=2yx2

Or you can write the answer as a product, repeat the base and subtract exponents.2x2y2x4y=2x2x4y2y1=2x2y1=2yx2In the last step, x2 is the same as 1/x2, then we multiply numerators and denominators.

Note that the right-hand method is more efficient.

Exercise 7.3.2

Simplify: 3yx3÷y24x.

Answer

12x2y

Adding and Subtracting Rational Expressions

First, recall the rules for adding or subtracting fractions that have a “common” denominator.

Adding rational expressions

Given a/c and b/c, their sum is defined as:ac+bc=a+bcThat is, add the numerators and place the result over the common denominator.

The following examples each share a common denominator. We add the numerators, then place the result over the common denominator.

57+17=67,2x+3x=5x,andxy+3yy=x+3yy

Example 7.3.3

Simplify: 3xxy+2yxy.

Solution

Add the numerators, placing the result over the common denominator.

3xxy+2yxy=3x+2yxy

Exercise 7.3.3

Simplify: 4xx2y+5y2x2y

Answer

4x+5y2x2y

Subtracting rational expressions

Given a/c and b/c, their difference is defined as:acbc=abcThat is, subtract the numerators and place the result over the common denominator.

The following examples each share a common denominator. We subtract the numerators, then place the result over the common denominator.

7929=29,5ab3ab=2ab,and3xxy5yxy=3x5yxy

As the next example shows, sometimes you may have to reduce your answer to lowest terms.

Example 7.3.4

Simplify: 5xy2z3xy2z.

Solution

Subtract the numerators, placing the result over the common denominator.

5xy2z3xy2z=5xy3xy2z=2xy2z

To reduce to lowest terms, divide both numerator and denominator by 2.

xyz

Exercise 7.3.4

Simplify: 8x3yz22x3yz2.

Answer

2xyz2

The Least Common Denominator

When adding or subtracting, if the rational expressions do not share a common denominator, you must first make equivalent fractions with a common denominator.

Least common denominator

If the fractions a/b and c/d do not share a common denominator, then the least common denominator for b and d is defined as the smallest number (or expression) divisible by both b and d. In symbols, LCD(b,d) represents the least common denominator of b and d.

Example 7.3.5

Simplify: x6+2x9.

Solution

The smallest number divisible by both 6 and 9 is 18; i.e., LCD(6,9)=18. We must first make equivalent fractions with a common denominator of 18.

x6+2x9=x633+2x922=3x18+4x18

=7x18

Exercise 7.3.5

Simplify: 3x8+5x6.

Answer

29x24

Example 7.3.6

Simplify: y8xy12x.

Solution

The smallest expression divisible by both 8x and 12x is 24x; i.e., LCD(8x,12x)=24x. We must first make equivalent fractions with a common denominator of 24x, then place the difference of the numerators over the common denominator.

y8xy12x=y8x33y12x22=3y24x2y24x=y24x

Exercise 7.3.6

Simplify: x8yx10y.

Answer

x40y

In Example 7.3.5, it was not difficult to imagine the smallest number divisible by both 6 and 9. A similar statement might apply to Example 7.3.6. This is not the case in all situations.

Example 7.3.7

Simplify: 5y72y108.

Solution

In this example, it is not easy to conjure up the smallest number divisible by both 72 and 108. As we shall see, prime factorization will come to the rescue.

fig 7.3.a.png

Thus, 72=2332 and 108=2233.

Note: Procedure for finding the least common denominator (LCD)

To find the least common denominator for two or more fractions, proceed as follows:

  1. Prime factor each denominator, putting your answers in exponential form.
  2. To determine the LCD, write down each factor that appears in your prime factorizations to the highest power that it appears.

Following the procedure above, we list the prime factorization of each denominator in exponential form. The highest power of 2 that appears is 23. The highest power of 3 that appears is 33.

72=2332Prime factor 72.108=2233Prime factor 108.LCD=2333Highest power of 2 is 23. Highest power of 3 is 33.

Therefore, the LCD is 2333=827 or 216. Hence:

5y72y108=5y7233y10822Make equivalent fractions.=15y2162y216Simplify.=13y216Subtract numerators.

Exercise 7.3.7

Simplify: 7x363x40.

Answer

43x360

Example 7.3.8

Simplify: 715xy21120x2

Solution

Prime factor each denominator, placing the results in exponential form.

15xy2=35xy220x2=225x2

To find the LCD, list each factor that appears to the highest power that it appears.

LCD=2235x2y2

Simplify.

LCD=60x2y2

After making equivalent fractions, place the difference of the numerators over this common denominator.

715xy21120x2=715xy24x4x1120x23y23y2=28x60x2y233y260x2y2=28x33y260x2y2

Exercise 7.3.8

Simplify: 1118xy2+7x30xy

Answer

55+21x290x2y

Dividing a Polynomial by a Monomial

We know that multiplication is distributive with respect to addition; that is, a(b+c)=ab+ac. We use this property to perform multiplications such as: x2(2x23x8)=2x43x38x2 However, it is also true that division is distributive with respect to addition.

Distributive property for division

If a, b, and c are any numbers, then:a+bc=ac+bc

For example, note that4+62=42+62

This form of the distributive property can be used to divide a polynomial by a monomial.

Example 7.3.9

Divide x22x3 by x2.

Solution

We use the distributive property, dividing each term by x2.

x22x3x2=x2x22xx23x2

Now we reduce each term of the last result to lowest terms, canceling common factors.

=12x3x2

Exercise 7.3.9

Divide 9x3+8x26x by 3x2.

Answer

3x+832x

Example 7.3.10

Divide 2x33x+12 by 6x3.

Solution

We use the distributive property, dividing each term by 6x3.

2x33x+126x3=2x36x33x6x3+126x3

Now we reduce each term of the last result to lowest terms, canceling common factors.

=1312x2+2x3

Exercise 7.3.10

Divide 4x2+6x9 by 2x4.

Answer

2x2+3x39x4


This page titled 7.3: Simplifying Rational Expressions is shared under a CC BY-NC-ND 3.0 license and was authored, remixed, and/or curated by David Arnold via source content that was edited to the style and standards of the LibreTexts platform.

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