LEMMA

Rational expressions

Simplify rational expressions

ALEKS placement

Factor top and bottom completely, then cancel matching factors — never terms.

What this covers

  • Factor the numerator and denominator fully. List the inputs that make the original denominator zero before canceling anything.
  • Write top and bottom as chains of multiplied pieces. Cross out a piece only if it multiplies the whole side, never if it is added.
  • Cancel factors that appear in both. Opposite-order factors like (5 − x) and (x − 5) cancel to −1, not 1.
  • State the restrictions: any value that made the original denominator zero.
  • Try an allowed input in the original and the simplified expression. Their values should match; factoring and legal cancellation justify the identity.
  • Leave the result factored or expanded, whichever was asked for.

Worked example

Worked example

Simplify (x² − 9)/(x² + x − 6)

  1. x2+x−6=(x+3)(x−2)Factor the denominator first. Its two factors show where the original fraction is undefined.
  2. x=−3x=2​Neither denominator factor may be zero. Keep both restrictions for every later line.
  3. x2−9=(x+3)(x−3)The numerator is a difference of squares. Factor the entire top before looking for a shared factor.
  4. (x+3)(x−2)(x+3)(x−3)​Now the same whole factor multiplies both the numerator and denominator.
  5. x−2x−3​The shared factor divided by itself equals one on the allowed domain. Cancel that factor, keeping both original restrictions.
  6. x=−3x=2​The simplified formula still represents only the original allowed inputs. Canceling a factor does not restore its missing input.
Another worked example

Simplify (6x² + 3x)/(3x)

  1. x=0The original denominator is zero at a zero input. Exclude that input before simplifying.
  2. 3x3x(2x+1)​Factor the entire numerator. The second term leaves one inside the group.
  3. 2x+1Cancel the shared multiplying factor on the allowed domain.
  4. x=0Keep the original restriction even though the reduced formula has no variable denominator.

A common mistake

A mistake Lemma catches

From (x+3)(x−2)(x+3)(x−3)​

(x+3)(x−2)(x+3)(x−3)​=−2−3​(x+3)(x−2)(x+3)(x−3)​=x−2x−3​

“Only whole factors cancel. (x + 3) multiplies the top and the bottom, so it cancels; the x inside x − 3 and x − 2 is part of a sum, not a factor, so it cannot. Test x = 5: (5 − 3)/(5 − 2) = 2/3, but −3/−2 = 3/2.”

Try one

Sample problem

Simplify and include the excluded value.

x−5x2−2x−15​

Practice rational simplify free

No account, no email. Every line you type is checked by the same computer algebra system as the worked example above.

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