Guides / Rational expressions

Simplifying rational expressions: cancel factors, not terms

Almost every wrong answer on a rational-expression problem comes from one line: the line where something gets crossed out that was never a factor. Here is how to tell the difference before you pick up the pencil.

Short answer

Factor the numerator and the denominator completely, then cancel only factors that appear on both top and bottom. A factor is something the whole numerator (or denominator) is multiplied by, like (x + 3) in (x − 3)(x + 3). A term is something added or subtracted, like the x in x + 6, and terms never cancel. Before you cancel anything, write down the values that make the original denominator zero. They stay excluded even after the factor that caused them is gone.

Why it works

Cancelling is dividing the top and bottom by the same thing. That only works when the thing divides all of the top and all of the bottom:

a · cb · c = ab · cc = abc/c is 1, as long as c isn't 0. This is the only move "cancelling" is allowed to make.
4 + 64 + 3 = 107Crossing out the 4s would give 6/3 = 2. But 10/7 isn't 2. The 4 is added, not multiplied, so there is no c/c to remove.

Put x where the 4 is and you have the most common mistake in this topic: (x + 6)/(x + 3) is not 2. It is already in simplest form. Plug in any number to see it: at x = 1 it is 7/4.

Example 1: factor both, cancel the shared factor

Simplify x² − 9x² + 5x + 6.

(x − 3)(x + 3)(x + 2)(x + 3)Difference of squares on top. On the bottom, two numbers that multiply to 6 and add to 5: 2 and 3.
x ≠ −2, x ≠ −3Read the excluded values off the factored denominator now, before anything is cancelled.
x − 3x + 2, x ≠ −2, −3(x + 3) multiplies the whole top and the whole bottom, so it cancels.

Notice the x²s did not cancel and the 9 and 6 did not reduce to 3/2. They were terms. Only the factor (x + 3) came out.

Example 2: pull out the common factor first

Simplify 2x² − 8xx² − 16.

2x(x − 4)(x − 4)(x + 4)Take out 2x on top. "Factor completely" starts with the greatest common factor, then looks inside the parentheses.
2xx + 4, x ≠ 4, −4Cancel (x − 4). The x on top and the x in x + 4 can't cancel: the bottom x is a term.

Example 3: factors that are opposites

Simplify 6 − 2xx² − 9.

−2(x − 3)(x − 3)(x + 3)6 − 2x is −2(x − 3). Factor out −2, not 2, so the binomial matches the one on the bottom. Check: −2 · x = −2x and −2 · (−3) = +6.
−2x + 3, x ≠ 3, −3Cancel (x − 3). The minus sign stays.

The shortcut version: (a − b)/(b − a) = −1, because the top and bottom are the same number with opposite signs. If you factor out 2 instead of −2 you get 2(3 − x), which doesn't match (x − 3), and that mismatch is your cue to pull out the negative.

Common mistakes, and the exact line they happen on

1. Cancelling terms

x + 6x + 3Already in simplest form. Neither the top nor the bottom factors any further.
63 = 2This is the mistake. The x's are added to the 6 and the 3, not multiplied, so they can't be divided out. At x = 1 the original is 7/4, not 2.

Test before you cross out: can you draw a box around the thing so that the rest of the numerator is multiplied by it? If there is a + or − sign outside the box, it's a term.

2. Cancelling from only one term of the numerator

x² + 5xxThe x on the bottom has to divide the whole top.
x² + 5This is the mistake. The x was cancelled out of the 5x only. Factor first: x(x + 5)/x = x + 5, for x ≠ 0. Both terms lose an x.

3. Treating opposites as equal

x − 55 − x = 1This is the mistake. 5 − x is −(x − 5), so the quotient is −1. Try x = 7: 2/(−2) = −1.

4. Dropping the excluded values

x² − 9x − 3 = (x − 3)(x + 3)x − 3Factor.
= x + 3This is the incomplete line. The simplified form is x + 3 for x ≠ 3. At x = 3 the original is 0/0, undefined, while x + 3 would say 6. When you later solve an equation with rational expressions, an answer of x = 3 has to be thrown out, and this note is how you know.

5. Cancelling before factoring completely

In (2x² + 7x + 3)/(4x² − 1), nothing visible matches until both sides are factored: (2x + 1)(x + 3) over (2x − 1)(2x + 1). If you stop factoring early, you either cancel something illegal or decide nothing cancels when (2x + 1) does. Factor everything, then look.

Practice

Simplify (x² − 25)/(x² − 3x − 10).
Show answer
(x − 5)(x + 5)/((x − 5)(x + 2)) = (x + 5)/(x + 2), for x ≠ 5, −2.
Simplify (3x + 12)/(x² + 4x).
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3(x + 4)/(x(x + 4)) = 3/x, for x ≠ 0, −4.
Simplify (4 − x)/(x² − 16).
Show answer
4 − x = −(x − 4), so −(x − 4)/((x − 4)(x + 4)) = −1/(x + 4), for x ≠ 4, −4.
A classmate writes (x + 8)/(x + 2) = 4. What is the simplified form?
Show answer
It is already simplified: (x + 8)/(x + 2), for x ≠ −2. The 8 and 2 are terms, so nothing cancels. At x = 1 the expression is 9/3 = 3, not 4.
Simplify (2x² + 7x + 3)/(4x² − 1).
Show answer
(2x + 1)(x + 3)/((2x − 1)(2x + 1)) = (x + 3)/(2x − 1), for x ≠ 1/2, −1/2.