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## 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.

x^{2} / x^{2} −9 is an example of a rational expression.

Since the denominator cannot equal 0, x^{2} −9 ≠0

x2 ≠9 → x≠3 or −3

So, the domain of x^{2} / x^{2} −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

x^{3}−5x^{2}−24x / x^{3}+x^{2}−72x

**Step 1: **

Factor the numerator and the denominator.

x^{3}−5x^{2}−24x / x^{3}+x^{2}−72x = x(x^{2}−5x−24) / x(x^{2}+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

x^{3}−5x^{2}−24x / x^{3}+x^{2}−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−x^{2} / x^{2}+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−x^{2} / x^{2}+3x−10 is −x+2 / x+5 for all real numbers except 2 and -5.

### Multiply and Divide Rational Expressions

#### Multiply Rational Expressions

**What is the product of**2xy / z and 3x^{2}/ 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 3x^{2} / 4yz is 3x^{3} / 2z^{2} for y ≠0 and z ≠0.

**What is the simplified form of the expression given below:**

#### Multiply a Rational Expression by a Polynomial

**What is the product of x+2 / x ^{4}−16 and x^{3} + 4x^{2} – 12x?**

#### Divide Rational Expressions

**What is the quotient of ** x^{3}+3x^{2}+3x+1 / 1−x^{2} and x^{2}+5x+4 / x^{2}+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: 2πx ^{2} + 2πx(2x)**

**Volume:** πx^{2} (2x)

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

**Option 1:**

SA/V = 2(4x^{2})+4(4x^{2}) / 8x^{3}

= 24x^{2} / 8x^{3} = 3x

**Option 2:**

SA/V = 2πx^{2}+4πx^{2} / 2πx^{3}

= 6πx^{2} / 2πx^{3} = 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. **

x^{3}+4x^{2}−x−4/x^{2}+3x−4

**Solution:**

x^{3}+4x^{2}−x−4/x^{2}+3x−4

= x^{2}(x+4)−1(x+4) / x^{2}+4x−x−4

= (x^{2}−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. **

- (x
^{2}−16)/ (9−x) ⋅ x^{2}+x−90 / x^{2}+14x+40

- x
^{3}−4x / 6x^{2}−13x−5 ⋅ (2x^{3}−3x^{2}−5x)

**Solution: **

1. **(x ^{2}−16)/ (9−x) ⋅ x^{2}+x−90 / x^{2}+14x+40 **

= (x2−16)/(9−x) ⋅ (x^{2}+10x−9x−90)/(x^{2}+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. x ^{3}−4x / 6x^{2}−13x−5 ⋅ (2x^{3}−3x^{2}−5x)**

= x(x^{2}−4)/6x^{2}+2x−15x−5 ⋅ x(2x^{2}−3x−5)

= x(x^{2}−4)/2x(3x+1)−5(3x+1) ⋅ x(2x^{2}+2x−5x−5)

= x(x^{2}−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)

= x^{2}(x−2)(x+2)(2x−5)(x+1)/(2x−5)(3x+1)

= x^{2}(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/x
^{2}+9x ÷ (6−x/3x^{2}−18x)

- 2x
^{2}−12x/x+5 ÷ (x−6/x+5)

**Solution: **

**1. 1/x ^{2}+9x ÷ (6−x/3x^{2}−18x)**

= 1/x^{2}+9x × 3x^{2}−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. 2x ^{2}−12x/x+5 ÷ (x−6/x+5)**

= 2x^{2}−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.

- 2a
^{2}b / b^{2}c . b/a - y
^{2}-2y-15/4 . 8/y+3 - x-5/6 ÷ 2x-10/12
- 5n+15/4n+8 . 2n+4/3n+9
- x
^{2}-2x ÷ 3x-6/x - m
^{2}-2m-8/8m+24 ÷ 2m-8/m^{2}+7m+12 - x+3/10x+20 . x+2/x
^{2}+4x+3 - x
^{2}-x-12/x-4 ÷ 2x+6/x-5 - x
^{2}-5x-6/5x+15 ÷ x^{2}-3x-4/7x+21 - 24x
^{3}/25y^{5}. 15y^{2}/8x^{2}

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