Simplifying Rational Expressions

MATH 018

Combined Algebra

S. Rook

2

Overview

• Section 7.1 in the textbook:– Evaluate a rational expression– Domain of rational expressions

• Find where a rational expression is undefined

– Simplify rational expressions– Simplify equivalent expressions

Evaluating a Rational Expression

4

Evaluating a Rational Expression

• Rational Expression: an expression of the form P/Q where P and Q are polynomials and Q ≠ 0

• To evaluate a rational expression:– Substitute the given values for the variable(s)

in the rational expression– Simplify the final answer!

Evaluating a Rational Expression (Example)

Ex 1: Evaluate for the given value of x and simplify:

a) ; when x = 6

b) ; when x = -2

5

76

52

xx

x

23

1

xx

x

6

Domain of Rational Expressions

77

Domain of Rational Expressions

• Domain: set of allowable values for x

• The domain can also mean those values where the rational expression is UNDEFINED

• A rational expression can be viewed as a fraction– When is a fraction undefined?

• Basically an exercise in factoring

Domain of Rational Expressions

Ex 2: Find the values where the rational expression is undefined:

a)

b)

8

932

112

xx

x

yy

y

81

173

2

9

Simplifying Rational Expressions

1010

Simplifying Rational Expressions

• Consider simplifying 20 / 30 using prime factorization

• Works the same way with rational expressions– i.e. completely factoring a polynomial is the

equivalent of prime factorization– Factor the numerator and denominator– Divide out common factors

• For polynomials other than monomials, common factors MUST have the SAME terms AND the SAME signs

Simplifying Rational Expressions (Continued)

• When dealing with a polynomial OTHER than a monomial:– Either divide out EVERYTHING

– Or divide out NOTHING AT ALL

• This is a common mistake!!!11

12

2

x

x

12

2

2

2

x

x

x

x

Simplifying Rational Expressions (Example)

Ex 3: Simplify:

a)

b)

12

8

24633

2

x

xx

55

10102

4

x

x

Simplifying Equivalent Expressions

14

Simplifying Equivalent Expressions

• Always be on the lookout for EQUIVALENT FACTORS (same factors just written in a different form) in a rational expression– Cleans up the final answer– Makes the process of adding or subtracting

rational expressions much easier

• Recall the commutative property of additionx + 7 and 7 + x are equivalent

– Thus, what does (x + 7)/(7 + x) simplify to?

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Simplifying Equivalent Expressions (Continued)

• Factoring out a negative SOMETIMES results in two equivalent factors:

– What would (x – 5)/(5 – x) simplify to?

– What about (x + 5)/(x – 5)?

Simplifying Equivalent Expressions (Example)

Ex 4: Simplify completely:

a)

b)

c)

16

x

xx

1025

572 2

364

182

x

x

202

10

x

x

17

Summary

• After studying these slides, you should know how to do the following:– Evaluate a rational expression– Determine where a rational expression is undefined– Simplify a rational expression– Recognize and simplify equivalent forms of a rational

expression• Additional Practice

– See the list of suggested problems for 7.1• Next lesson

– Multiplying & Dividing Rational Expressions (Section 7.2)