Virginia SOL Math / Algebra II / Expressions & Operations

A2.EO.1

Simplify Rational Expressions

What you'll learn

  • Factor numerator and denominator completely
  • Simplify a rational expression by dividing out common factors
  • State the values of the variable that must be excluded

Words to know

Rational expression —
a ratio of two polynomials
Factor —
write a polynomial as a product of simpler polynomials
Common factor —
a factor shared by numerator and denominator
Excluded value —
a value that makes the denominator 0
Simplest form —
no common factors remain except 1

Why it matters Simplifying rational expressions makes them easier to evaluate, combine, and solve, the same way reducing a fraction makes it easier to work with.

Worked examples

Model ItI do

Simplify (x^2 - 9) / (x^2 + x - 6).

Factor both parts: the top is (x - 3)(x + 3) and the bottom is (x - 3)(x + 2). Divide out the common factor (x - 3) to get (x + 3) / (x + 2). Exclude x = 3 and x = -2, since those make the original denominator 0.

Try It TogetherWe do

Simplify (2x + 6) / (x^2 - 9).

Factor: the top is 2(x + 3) and the bottom is (x - 3)(x + 3). Divide out (x + 3) to get 2 / (x - 3), with x not equal to 3 or -3.

Now YouYou do

Simplify (x^2 - 4x) / (x^2 - 16).

Factor the top and bottom, then divide out the common factor. Check your answer in the back of this chapter.

Practice

Mastered 0 of 11

Warm-up

Ease in with the basics.

  1. 1.Simplify (x^2 - 4) / (x + 2).

  2. 2.What value must be excluded in 8 / (x - 3)?

  3. 3.Simplify (5x + 10) / 5.

On Level

The core Quick Check from the book.

  1. 1.Simplify (x^2 - 25) / (x + 5).

  2. 2.Simplify (3x + 12) / (x^2 - 16).

  3. 3.What value must be excluded in 5 / (x - 7)?

  4. 4.Simplify (x^2 + 2x) / x.

  5. 5.Simplify (x^2 - x - 6) / (x^2 - 9).

Challenge

Stretch your thinking.

  1. 1.Simplify (x^2 - x - 12) / (x^2 - 9).

  2. 2.Simplify (2x^2 + 7x + 3) / (x^2 + 6x + 9).

  3. 3.Simplify (x^2 - 5x + 6) / (x^2 - 4).