Multiply and Divide

Rational ExpressionsLESSON 6.1

Objective

Determine the domain of rational

expressions and functions

Simplify, multiply, and divide rational

expressions and functions

Rational Expressions

A rational expression is the quotient (÷) of

two polynomials

EX: 𝑥−5

2𝑥+1

𝑥2−7𝑥−18

𝑥2−4

1

𝑥−3

2𝑎2+5𝑎𝑏+2𝑏2

𝑎2−6𝑎𝑏+8𝑏2

Rational Expressions - Domain

REMEMBER: We cannot ÷0

Domain consists of all real numbers

except those for which the denominator

is zero.

Rational Expressions - Domain

Determine the domain for each of the

following rational expressions

A. −3𝑧

𝑧+5B.

𝑛2−2𝑛−8

𝑛2−𝑛−12

Rational Expressions - Domain

Determine the domain for each of the

following rational expressions

C. 𝑅 𝑥 =𝑥−3

𝑥2−2𝑥−8

Rational Expressions - Simplify

Rational expressions can be simplified by

dividing out any common factors

EX: 10

10= 1

𝑥

𝑥= 1

𝑥−3

𝑥−3= 1

2 𝑥−3

3(𝑥−3)=

2

3

Rational Expressions - Simplify

Simplify

D. 𝑥2+𝑥−6

2𝑥2−5𝑥+2E.

𝑦3+27

2𝑦2+6𝑦

Rational Expressions - Multiply

MULTIPLYING RATIONAL EXPRESSIONS

1. Completely factor each polynomial

2. Divide out common factors

3. Multiply remaining terms (fractions)

Rational Expressions - Multiply

F. 𝑛2−9

𝑛2+5𝑛+6×

𝑛+2

−2𝑛+6

Rational Expressions - Multiply

G. 𝑎2−𝑏2

10𝑎2−10𝑎𝑏×

10𝑎+5𝑏

2𝑎2+3𝑎𝑏+𝑏2

Rational Expressions - Divide

DIVIDING RATIONAL EXPRESSIONS𝑎𝑏𝑐𝑑

=𝑎

𝑏÷𝑐

𝑑=𝑎

𝑏×𝑑

𝑐

KFC→Keep the First, Flip the Second, Change the Sign

Rational Expressions - Divide

H.

45𝑧4

7𝑦5𝑧

21𝑦2

Rational Expressions - Divide

I.

𝑝3−8

5𝑝2+15𝑝

𝑝2−4

𝑝2+3𝑝

Rational Expressions - Functions

Sometimes we are given two or more

functions and told to combine and

simplify them.

EX: 𝑓 𝑥

𝑔 𝑥𝑓 𝑥 ⋅ 𝑔(𝑥)

Rational Expressions - Functions

J. 𝑓 𝑥 =𝑥2−9

2𝑥2−8𝑥𝑔 𝑥 =

𝑥−4

𝑥2+4𝑥+3ℎ 𝑥 =

𝑥2+6𝑥+9

𝑥2−5𝑥

1. Find 𝑓 𝑥 ⋅ 𝑔(𝑥)

2. Find 𝑓 𝑥

ℎ 𝑥