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.
- x+23x
- x+3x2−2x−4
- 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:ab⋅cd=acbd
Remember, you need only multiply numerators and denominators. For example:
- x3⋅2y=2x3y
- 2a3b2⋅5a9b3=10a227b5
- 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: 2x⋅x24.
Solution
Multiply numerators and denominators.
2x⋅x24=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=2⋅x⋅x⋅x⋅x⋅x2⋅2⋅x⋅x⋅x=⧸2⋅⧸x⋅⧸x⋅⧸x⋅x⋅x⧸2⋅2⋅⧸x⋅⧸x⋅⧸x=x22
Or you can write the answer as a product, repeat the base and subtract exponents.2x54x3=24⋅x5x3=12⋅x5−3=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: 9x2⋅x6.
- 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=ab⋅dc=adbc
Example 7.3.2
Simplify: x2y÷x42y2.
Solution
Invert, then multiply.
x2y÷x42y2=x2y⋅2y2x4=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=2⋅x⋅x⋅y⋅yx⋅x⋅x⋅x⋅y=2⋅⧸x⋅⧸x⋅⧸y⋅y⧸x⋅⧸x⋅x⋅x⋅⧸y=2yx2
Or you can write the answer as a product, repeat the base and subtract exponents.2x2y2x4y=2⋅x2x4⋅y2y1=2x−2y1=2yx2In the last step, x−2 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:ac−bc=a−bcThat 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.
79−29=29,5ab−3ab=2ab,and3xxy−5yxy=3x−5yxy
As the next example shows, sometimes you may have to reduce your answer to lowest terms.
Example 7.3.4
Simplify: 5xy2z−3xy2z.
Solution
Subtract the numerators, placing the result over the common denominator.
5xy2z−3xy2z=5xy−3xy2z=2xy2z
To reduce to lowest terms, divide both numerator and denominator by 2.
xyz
Exercise 7.3.4
Simplify: 8x3yz2−2x3yz2.
- 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=x6⋅33+2x9⋅22=3x18+4x18
=7x18
Exercise 7.3.5
Simplify: 3x8+5x6.
- Answer
-
29x24
Example 7.3.6
Simplify: y8x−y12x.
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.
y8x−y12x=y8x⋅33−y12x⋅22=3y24x−2y24x=y24x
Exercise 7.3.6
Simplify: x8y−x10y.
- 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: 5y72−y108.
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.
Thus, 72=23⋅32 and 108=22⋅33.
Note: Procedure for finding the least common denominator (LCD)
To find the least common denominator for two or more fractions, proceed as follows:
- Prime factor each denominator, putting your answers in exponential form.
- 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=23⋅32Prime factor 72.108=22⋅33Prime factor 108.LCD=23⋅33Highest power of 2 is 23. Highest power of 3 is 33.
Therefore, the LCD is 23⋅33=8⋅27 or 216. Hence:
5y72−y108=5y72⋅33−y108⋅22Make equivalent fractions.=15y216−2y216Simplify.=13y216Subtract numerators.
Exercise 7.3.7
Simplify: 7x36−3x40.
- Answer
-
43x360
Example 7.3.8
Simplify: 715xy2−1120x2
Solution
Prime factor each denominator, placing the results in exponential form.
15xy2=3⋅5⋅x⋅y220x2=22⋅5⋅x2
To find the LCD, list each factor that appears to the highest power that it appears.
LCD=22⋅3⋅5⋅x2⋅y2
Simplify.
LCD=60x2y2
After making equivalent fractions, place the difference of the numerators over this common denominator.
715xy2−1120x2=715xy2⋅4x4x−1120x2⋅3y23y2=28x60x2y2−33y260x2y2=28x−33y260x2y2
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(2x2−3x−8)=2x4−3x3−8x2 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 x2−2x−3 by x2.
Solution
We use the distributive property, dividing each term by x2.
x2−2x−3x2=x2x2−2xx2−3x2
Now we reduce each term of the last result to lowest terms, canceling common factors.
=1−2x−3x2
Exercise 7.3.9
Divide 9x3+8x2−6x by 3x2.
- Answer
-
3x+83−2x
Example 7.3.10
Divide 2x3−3x+12 by 6x3.
Solution
We use the distributive property, dividing each term by 6x3.
2x3−3x+126x3=2x36x3−3x6x3+126x3
Now we reduce each term of the last result to lowest terms, canceling common factors.
=13−12x2+2x3
Exercise 7.3.10
Divide −4x2+6x−9 by 2x4.
- Answer
-
−2x2+3x3−9x4