Operations with rational expressions

Brian Hall

7 min read

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Study Guide Overview

This guide covers rational expressions for the AP SAT (Digital). It reviews simplifying rational expressions through factoring, long division, and splitting fractions. It also explains operations on rational expressions, including addition, subtraction, multiplication, division, and simplifying complex fractions. Key factoring techniques like GCF, grouping, difference of squares, and sum/difference of cubes are highlighted. The guide includes practice questions and emphasizes common pitfalls and strategies for success.

#AP SAT (Digital) Prep: Rational Expressions - Your Night-Before Guide 🚀

Hey there! Let's get you prepped for those rational expressions. Think of this as your quick-hit guide to make sure you're feeling confident and ready to ace this section. We're breaking it down, making it simple, and getting you exam-ready. Let's do this!

#Rational Expressions: The Big Picture

Rational expressions are essentially fractions with polynomials. They might seem intimidating, but they're just a combination of things you already know: factoring, fractions, and a little bit of algebraic manipulation. You've got this! We'll cover simplifying, adding, subtracting, multiplying, and dividing. Let's jump in!

# Simplifying Rational Expressions

#Factoring and Canceling Common Factors

What are they? Rational expressions are fractions where the numerator and denominator are polynomials.
The Goal: To reduce the expression to its simplest form.
How? Factor both the numerator and the denominator completely.
- Techniques:
  - Greatest Common Factor (GCF): Pull out the biggest common factor.
  - Grouping: Useful for four-term polynomials.
  - Difference of Squares: $a^2 - b^2 = (a+b)(a-b)$

Memory Aid

Think "Difference of Squares" as "Sum and Difference"

- **Sum or Difference of Cubes:**

a^3 + b^3 = (a+b)(a^2-ab+b^2)

and

a^3 - b^3 = (a-b)(a^2+ab+b^2)

Memory Aid

Remember the acronym SOAP: Same sign, Opposite sign, Always Positive for the second part of the expansion

- **Cancel:** Once factored, cancel out any common factors in the numerator and denominator.

Example: Simplify $\frac{x^2 - 4}{x^2 - 2x - 3}$

Factor: $\frac{(x+2)(x-2)}{(x-3)(x+1)}$
Cancel: No common factors to cancel in this case.
Simplified: $\frac{(x+2)(x...

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Give us your feedback and let us know how we can improve

Previous Topic - Operations with polynomials Next Topic - Nonlinear functions

Operations with rational expressions

Brian Hall

7 min read

Next Topic - Nonlinear functions

Listen to this study note

Study Guide Overview

#AP SAT (Digital) Prep: Rational Expressions - Your Night-Before Guide 🚀

#Rational Expressions: The Big Picture

# Simplifying Rational Expressions

#Factoring and Canceling Common Factors

What are they? Rational expressions are fractions where the numerator and denominator are polynomials.
The Goal: To reduce the expression to its simplest form.
How? Factor both the numerator and the denominator completely.
- Techniques:
  - Greatest Common Factor (GCF): Pull out the biggest common factor.
  - Grouping: Useful for four-term polynomials.
  - Difference of Squares: $a^2 - b^2 = (a+b)(a-b)$

Memory Aid

Think "Difference of Squares" as "Sum and Difference"

- **Sum or Difference of Cubes:**

a^3 + b^3 = (a+b)(a^2-ab+b^2)

and

a^3 - b^3 = (a-b)(a^2+ab+b^2)

Memory Aid

Remember the acronym SOAP: Same sign, Opposite sign, Always Positive for the second part of the expansion

- **Cancel:** Once factored, cancel out any common factors in the numerator and denominator.

Example: Simplify $\frac{x^2 - 4}{x^2 - 2x - 3}$

Factor: $\frac{(x+2)(x-2)}{(x-3)(x+1)}$
Cancel: No common factors to cancel in this case.
Simplified: $\frac{(x+2)(x...

How are we doing?

Give us your feedback and let us know how we can improve

Previous Topic - Operations with polynomials Next Topic - Nonlinear functions