How to Simplify Fractions with Variables

Here you will learn to simplify fractions that include variables. The most important part is factorizing the numerator and denominator, and then cancelling common factors.

Rule

Simplifying Fractions with Variables

1.
Factorize the numerator.
2.
Factorize the denominator.
3.
Cancel common factors.
4.
Multiply the remaining factors.

Example 1

Simplify the expression 4x + 16 4

Find the common factors and cancel:

4x + 16 4 = 4(x + 4) 4 = 4(x + 4) 4 = x + 4

4x + 16 4 = 4(x + 4) 4 = 4(x + 4) 4 = x + 4

Example 2

Simplify the expression 3x + 6 2x + 4

Find the common factors and cancel:

3x + 6 2x + 4 = 3(x + 2) 2(x + 2) = 3(x + 2) 2(x + 2) = 3 2

3x + 6 2x + 4 = 3(x + 2) 2(x + 2) = 3(x + 2) 2(x + 2) = 3 2

Example 3

Simplify the expression 3ab + 6a 2ab + 4a

Find the common factors and cancel:

3ab + 6a 2ab + 4a = 3a(b + 2) 2a(b + 2) = 3a(b + 2) 2a(b + 2) = 3 2

3ab + 6a 2ab + 4a = 3a(b + 2) 2a(b + 2) = 3a(b + 2) 2a(b + 2) = 3 2

Example 4

Simplify the expression 4x2 3x 4x 3

Find the common factors and cancel:

4x2 3x 4x 3 = x(4x 3) (4x 3) = x(4x 3) (4x 3) = x

4x2 3x 4x 3 = x(4x 3) (4x 3) = x(4x 3) (4x 3) = x

Example 5

Write x2 81 x 9 as simply as possible

Using the third algebraic identity of quadratic expressions, you get

x2 81 x 9 = (x 9)(x + 9) (x 9) = (x 9)(x + 9) (x 9) = x + 9

x2 81 x 9 = (x 9)(x + 9) (x 9) = (x 9)(x + 9) (x 9) = x + 9.

Think About This

Being able to factorize and cancel expressions is one of the main skills you need in mathematics. I recommend that you spend plenty of time practicing these skills.

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