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# Multiply and Divide Rational Expressions

## Key Concepts

• Write equivalent rational expressions.
• Simplify a rational expression.
• Multiply rational expressions.
• Multiply a rational expression by a polynomial.
• Divide rational expressions.

## Rational Expression

### Concept

A rational expression is the quotient of two polynomials. The domain is all real numbers except those for which the denominator is equal to 0.

x2 / x2 −9  is an example of a rational expression.

Since the denominator cannot equal 0, x2 −9 ≠0

x2 ≠9 → x≠3 or −3

So, the domain of x2 / x2 −9 is all real numbers except 3 and -3.

### Write Equivalent Rational Expressions

When are two rational expressions equivalent?

Rational expressions can be simplified in a process that is similar to the process for simplifying rational numbers.

12/16 = 3⋅2⋅2/2⋅2⋅2⋅2 = 3/2⋅2⋅2/2⋅2/2 = 3/2⋅2⋅1⋅1 = 3/4

By replacing quotients of common factors between the numerator and denominator with 1, you learn that

12/16 is equivalent to 3/4

Write an expression that is equivalent to

x3−5x2−24x / x3+x2−72x

Step 1:

Factor the numerator and the denominator.

x3−5x2−24x / x3+x2−72x = x(x2−5x−24) / x(x2+x−72) = x(x−8)(x+3) / x(x−8)(x+9)

Step 2:

Find the domain of the rational expression.

The domain is all real numbers except 0, 8, and -9.

Both

x3−5x2−24x / x3+x2−72x and x(x−8)(x+3) / x(x−8)(x+9) have the same domain.

Step 3:

Recognize the ratio of the common factors in the numerator and the denominator are equal to 1.

### Simplify a Rational Expression

What is the simplified form of the rational expression? What is the domain for which the identity between the two expressions is valid?

4−x2 / x2+3x−10

The simplified form of a rational expression has no common factors, other than 1, in the numerator and the denominator.

The simplified form of 4−x2 / x2+3x−10 is −x+2 / x+5 for all real numbers except 2 and -5.

### Multiply and Divide Rational Expressions

#### Multiply Rational Expressions

1. What is the product of 2xy / z and 3x2 / 4yz ?

To multiply rational expressions, follow a similar method to that for multiplying two numerical fractions. The domain is z ≠0 and y ≠0.

The product of 2xy / z and 3x2 / 4yz is 3x3 / 2z2 for y ≠0 and z ≠0.

1. What is the simplified form of the expression given below:

#### Multiply a Rational Expression by a Polynomial

What is the product of x+2 / x4−16 and x3 + 4x2 – 12x?

#### Divide Rational Expressions

What is the quotient of  x3+3x2+3x+1 / 1−x2 and x2+5x+4 / x2+3x−4?

The quotient is –(x + 1), x≠−4, −1, or 1.

#### Use Division of Rational Expressions

A company is evaluating two packaging options for its product line. The more efficient design will have a lesser ratio of surface area to volume. Should the company use packages that are cylinders or rectangular prisms?

Option 1: A rectangular prism with a square base

Option 2: A cylinder with the same height as the prism, and diameter equal to the side length of the prism’s base.

Surface Area: 2(2x)2 + 4(2x)2

Volume: (2x)3(𝟐𝒙)𝟑

Surface Area: 2πx2 + 2πx(2x)

Volume: πx2 (2x)

The efficiency ratio is SA/V, where SA represents Surface Area and V represents volume.

Option 1:

SA/V = 2(4x2)+4(4x2) / 8x3

= 24x2 / 8x3 = 3x

Option 2:

SA/V = 2πx2+4πx2 / 2πx3

= 6πx2 / 2πx3 = 3/x

The company can now compare the efficiency ratio of the package designs. Prism: 3/x and Cylinder:

3/x

In this example, the efficiency ratio of the cylinder is equal to that of the prism. So, the company should choose its package design based on other criteria.

### Questions

Question 1

Simplify the expression and state the domain.

x3+4x2−x−4/x2+3x−4

Solution:

x3+4x2−x−4/x2+3x−4

= x2(x+4)−1(x+4) / x2+4x−x−4

= (x2−1)(x+4) / x(x+4)−1(x+4)

= (x−1)(x+1)(x+4) / (x−1)(x+4)

= x+1 for x≠1,−4

Question 2

Find the simplified form of each product and give the domain.

1. (x2−16)/ (9−x) ⋅ x2+x−90 / x2+14x+40
1. x3−4x / 6x2−13x−5 ⋅ (2x3−3x2−5x)

Solution:

1. (x2−16)/ (9−x) ⋅ x2+x−90 / x2+14x+40

= (x2−16)/(9−x) ⋅ (x2+10x−9x−90)/(x2+10x+4x+40)

= (x2−16)/(9−x) ⋅ (x(x+10)−9(x+10))/(x(x+10)+4(x+10))

= (x−4)(x+4)/−(x−9) ⋅ (x−9)(x+10)/(x+4)(x+10)

= (x−4)(x+4)/−(x−9) ⋅ (x−9)(x+10)/(x+4)(x+10)

= 4−x for x≠9,−4 or−10

2. x3−4x / 6x2−13x−5 ⋅ (2x3−3x2−5x)

= x(x2−4)/6x2+2x−15x−5 ⋅ x(2x2−3x−5)

= x(x2−4)/2x(3x+1)−5(3x+1) ⋅ x(2x2+2x−5x−5)

= x(x2−4)/(2x−5)(3x+1) ⋅ x(2x(x+1)−5(x+1))

= x(x−2)(x+2)/(2x−5)(3x+1) ⋅ x(2x−5)(x+1)

= x2(x−2)(x+2)(2x−5)(x+1)/(2x−5)(3x+1)

= x2(x−2)(x+2)(x+1)/(3x+1) for x≠5/2,−1/3

Question 3

Find the simplified quotient and the domain of each expression.

1. 1/x2+9x ÷ (6−x/3x2−18x)
1. 2x2−12x/x+5 ÷ (x−6/x+5)

Solution:

1. 1/x2+9x ÷ (6−x/3x2−18x)

= 1/x2+9x × 3x2−18x/6−x

= 1/x(x+9) × 3x/(x−6)−(x−6)

= 1/x(x+9) × 3x(x−6)/−(x−6)

= −3/x+9 for x≠0,−9, or 6

2. 2x2−12x/x+5 ÷ (x−6/x+5)

= 2x2−12x/x+5 × (x+5)/(x−6)

= 2x(x−6)/ x+5 × (x+5)/(x−6)

= 2x for x≠−5 or 6

## Exercise

Multiply or divide the following rational expressions.

• 2a2b / b2c . b/a
• y2-2y-15/4 . 8/y+3
• x-5/6 ÷ 2x-10/12
• 5n+15/4n+8 . 2n+4/3n+9
• x2-2x ÷ 3x-6/x
• m2-2m-8/8m+24 ÷ 2m-8/m2+7m+12
• x+3/10x+20 . x+2/x2+4x+3
• x2-x-12/x-4 ÷ 2x+6/x-5
• x2-5x-6/5x+15 ÷ x2-3x-4/7x+21
• 24x3/25y5 . 15y2/8x2

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