Rational Expressions and Operations: Precalculus Study Notes

Rational Expressions and Operations

Introduction

Rational expressions are fractions in which the numerator and denominator are polynomials. Operations on rational expressions—such as multiplication and division—are fundamental skills in algebra and precalculus, and are essential for simplifying complex expressions and solving equations.

Multiplying and Dividing Rational Expressions

When multiplying or dividing rational expressions, the same rules as for numerical fractions apply, but with polynomials:

• Multiplication: Multiply the numerators together and the denominators together, then simplify.

• Division: Multiply by the reciprocal of the divisor (flip numerator and denominator of the second fraction), then simplify.

Key Steps for Simplifying Rational Expressions

1. Factor all numerators and denominators completely.

2. Cancel any common factors between numerators and denominators.

3. For division, rewrite as multiplication by the reciprocal.

4. Simplify the resulting expression.

Examples and Applications

Example 1: Multiplying Rational Expressions

Given:

$\frac{r-s}{r+s} \cdot \frac{r^2-s^2}{(r-s)^2}$

• Factor $r^2 - s^2$ as $(r-s)(r+s)$.

• Simplify: $\frac{r-s}{r+s} \cdot \frac{(r-s)(r+s)}{(r-s)^2}$

• Cancel common factors: $\frac{r-s}{r+s} \cdot \frac{(r-s)(r+s)}{(r-s)^2} = \frac{r+s}{r-s}$

Example 2: Dividing Rational Expressions

Given:

$\frac{m^2-n^2}{r+s} \div \frac{m-n}{r+s}$

• Rewrite division as multiplication by the reciprocal:

• $\frac{m^2-n^2}{r+s} \cdot \frac{r+s}{m-n}$

• Factor $m^2-n^2$ as $(m-n)(m+n)$.

• Simplify: $\frac{(m-n)(m+n)}{r+s} \cdot \frac{r+s}{m-n} = m+n$

Example 3: Multiplying Rational Expressions with Polynomials

Given:

$\frac{4x^2+9x+2}{x^2+x-2} \cdot \frac{x^2-1}{3x^2+x-2}$

• Factor all polynomials:

  • $4x^2+9x+2 = (4x+1)(x+2)$

  • $x^2+x-2 = (x+2)(x-1)$

  • $x^2-1 = (x+1)(x-1)$

  • $3x^2+x-2 = (3x-2)(x+1)$

• Substitute and simplify:

• $\frac{(4x+1)(x+2)}{(x+2)(x-1)} \cdot \frac{(x+1)(x-1)}{(3x-2)(x+1)}$

• Cancel common factors: $(x+2), (x-1), (x+1)$

• Final simplified form: $\frac{4x+1}{3x-2}$

Example 4: Dividing Rational Expressions with Variables

Given:

$\frac{a^2-a-2}{a^2-a-6} \div \frac{a^2-2a}{2a+a^2}$

• Factor all polynomials:

  • $a^2-a-2 = (a-2)(a+1)$

  • $a^2-a-6 = (a-3)(a+2)$

  • $a^2-2a = a(a-2)$

  • $2a+a^2 = a(a+2)$

• Rewrite division as multiplication by reciprocal:

• $\frac{(a-2)(a+1)}{(a-3)(a+2)} \cdot \frac{a(a+2)}{a(a-2)}$

• Cancel common factors: $a, (a-2), (a+2)$

• Final simplified form: $\frac{a+1}{a-3}$

Example 5: Division with Binomial Squares

Given:

$\frac{3x+12}{2x-8} \div \frac{(x+4)^2}{(x-4)^2}$

• Factor:

  • $3x+12 = 3(x+4)$

  • $2x-8 = 2(x-4)$

• Rewrite division as multiplication by reciprocal:

• $\frac{3(x+4)}{2(x-4)} \cdot \frac{(x-4)^2}{(x+4)^2}$

• Cancel common factors: $(x+4), (x-4)$

• Final simplified form: $\frac{3(x-4)}{2(x+4)}$

Example 6: Multiplying Rational Expressions with Multiple Variables

Given:

$\frac{x^2-y^2}{x^3-y^3} \cdot \frac{x^2+xy+y^2}{x^2+2xy+y^2}$

• Factor:

  • $x^2-y^2 = (x-y)(x+y)$

  • $x^3-y^3 = (x-y)(x^2+xy+y^2)$

  • $x^2+2xy+y^2 = (x+y)^2$

• Substitute and simplify:

• $\frac{(x-y)(x+y)}{(x-y)(x^2+xy+y^2)} \cdot \frac{x^2+xy+y^2}{(x+y)^2}$

• Cancel common factors: $(x-y), (x^2+xy+y^2), (x+y)$

• Final simplified form: $\frac{1}{x+y}$

Summary Table: Rational Expression Operations

Operation	Step 1	Step 2	Step 3
Multiplication	Factor all polynomials	Multiply numerators and denominators	Simplify and cancel common factors
Division	Factor all polynomials	Multiply by reciprocal of divisor	Simplify and cancel common factors

Additional info:

• These examples cover key skills for manipulating rational expressions, which are foundational for later topics such as rational functions, equations, and inequalities.

• Factoring is essential for simplification and for finding domain restrictions (values that make denominators zero).