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8.1: Reduce Rational Expressions

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Definition: Rational Expression

A rational expression is a ratio of two polynomials, i.e., a fraction where the numerator and denominator are polynomials.

Evaluate Rational Expressions

Example 8.1.1

Evaluate x24x2+6x+8 when x=6.

Solution

x24x2+6x+8Plug-n-chug x=6(6)24(6)2+6(6)+8Simplify each numerator and denominator3643636+8Simplify328Reduce4Evaluated value

Find Excluded Values of Rational Expressions

Rational expressions are special types of fractions, but still hold the same arithmetic properties. One property of fractions we recall is that the fraction is undefined when the denominator is zero.

Note

A rational expression is undefined where the denominator is zero.

Determine the excluded value(s) of a rational expression

Step 1. Set the denominator of the rational expression equal to zero.

Step 2. Solve the equation for the given variable.

Step 3. The values found in the previous step are the values excluded from the expression.

Example 8.1.2

Find the excluded value(s) of the expression: 3zz+5

Solution

Step 1. Set the denominator of the rational expression equal to zero: z+5=0

Step 2. Solve the equation for z: z+5=0z=5

Step 3. The values found in the previous step are the values excluded from the expression. Hence, the excluded value is z=5.

Example 8.1.3

Find the excluded value(s) of the expression: x213x2+5x

Solution

Step 1. Set the denominator of the rational expression equal to zero: 3x2+5x=0

Step 2. Solve the equation for x: 3x2+5x=0x(3x+5)=0x=0or3x+5=0x=0or3x=5x=0orx=53

Step 3. The values found in the previous step are the values excluded from the expression. Hence, the excluded values are x=0 and x=5.

Recall, the excluded values are values in which make the expression undefined. Hence, when evaluating rational expressions, we can evaluate the expressions for any values except the excluded values.

Note

The number zero was not widely accepted in mathematical thought around the world for many years. It was the Mayans of Central America who first used zero to aid in the use of their base-20 system as a place holder.

Reduce Rational Expressions with Monomials

Rational expressions are reduced, just as in arithmetic, even without knowing the value of the variable. When we reduce, we divide out common factors as we discussed with polynomial division with monomials. Now, we use factoring techniques and exponent properties to reduce rational expressions.

Reducing Rational Expressions

If P,Q,K are non-zero polynomials and PKQK is a rational expression, then PKQK=PQ

We call a rational expression irreducible if there are no more common factors among the numerator and denominator.

Example 8.1.4

Simplify: 15x4y225x2y6

Solution

Since the denominator is a monomial, then we reduce as usual and apply exponent rules:

15x4y225x2y6Reduce by applying exponent rules3x25y4Reduced expression

Reduce Rational Expressions with Polynomials

However, if there is a sum or difference in either the numerator or denominator, we first factor the numerator and denominator to obtain a product of factors, then reduce.

Example 8.1.5

Simplify: 288x216

Solution

Since we have a difference in the denominator, we factor the denominator and then reduce.

288x216Factor a GCF 8 from the denominator4724(x22)Reduce by a factor of 472(x22)Reduced expression

Example 8.1.6

Simplify: 9x318x6

Solution

Since we have a difference in the denominator and numerator, we factor the denominator and numerator, and then reduce.

9x318x6Factor the GCF from numerator and denominator3(3x1)6(3x1)Reduce by a factor of 3(3x1)3(3x1)23(3x1)Rewrite the expression12Reduced expression

Example 8.1.7

Simplify: x225x2+8x+15

Solution

Since we have a sum and difference of terms in the denominator and numerator, we factor the denominator and numerator, and then reduce.

x225x2+8x+15Factor using factoring techniques(x+5)(x5)(x+3)(x+5)Reduce by a factor of (x+5)(x+5)(x5)(x+3)(x+5)Rewrite the expressionx5x+3Reduced expression

Note

We cannot reduce terms, only factors. This means we cannot reduce anything with a + or between the parts. In Example 8.1.7 , we obtained the reduced expression x5x+3. Note, we are not allowed to divide out the x’s because they are terms (separated by + or ) not factors (separated by multiplication).

Reduce Rational Expressions Homework

Evaluate the expression for the given value.

Exercise 8.1.1

4v+26 when v=4

Exercise 8.1.2

x3x24x+3 when x=4

Exercise 8.1.3

b+2b2+4b+4 when b=0

Exercise 8.1.4

b33b9 when b=2

Exercise 8.1.5

a+2a2+3a+2 when a=1

Exercise 8.1.6

n2n6n3 when n=4

Find the excluded value(s).

Exercise 8.1.7

3k2+30kk+10

Exercise 8.1.8

15n210n+25

Exercise 8.1.9

10m2+8m10m

Exercise 8.1.10

r2+3r+25r+10

Exercise 8.1.11

b2+12b+32b2+4b32

Exercise 8.1.12

27p18p236p

Exercise 8.1.13

x+108x2+80x

Exercise 8.1.14

10x+166x+20

Exercise 8.1.15

6n221n6n2+3n

Simplify each expression.

Exercise 8.1.16

21x218x

Exercise 8.1.17

24a40a2

Exercise 8.1.18

32x38x4

Exercise 8.1.19

18m2460

Exercise 8.1.20

204p+2

Exercise 8.1.21

x+1x2+8x+7

Exercise 8.1.22

32x228x2+28x

Exercise 8.1.23

n2+4n12n27n+10

Exercise 8.1.24

9v+54v24v60

Exercise 8.1.25

12x242x30x242x

Exercise 8.1.26

6a1010a+4

Exercise 8.1.27

2n2+19n109n+90

Exercise 8.1.28

21k24k2

Exercise 8.1.29

90x220x

Exercise 8.1.30

1081n3+36n2

Exercise 8.1.31

n99n81

Exercise 8.1.32

28m+1236

Exercise 8.1.33

49r+5656r

Exercise 8.1.34

b2+14b+48b2+15b+56

Exercise 8.1.35

30x9050x+40

Exercise 8.1.36

k212k+32k264

Exercise 8.1.37

9p+18p2+4p+4

Exercise 8.1.38

3x229x+405x230x80

Exercise 8.1.39

8m+1620m12

Exercise 8.1.40

2x210x+83x27x+4

Exercise 8.1.41

7n232n+164n16

Exercise 8.1.42

n22n+16n+6

Exercise 8.1.43

7a226a456a234a+20

Exercise 8.1.44

56x4824x2+56x+32

Exercise 8.1.45

50b8050b+20

Exercise 8.1.46

35v+3521v+7

Exercise 8.1.47

56x4824x2+56x+32

Exercise 8.1.48

4k32k22k9k318k2+9k


This page titled 8.1: Reduce Rational Expressions is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Darlene Diaz (ASCCC Open Educational Resources Initiative) via source content that was edited to the style and standards of the LibreTexts platform.

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