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.

2. 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 

2. 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)

2. 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

Key Concepts Covered  

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