Simplifying algebraic fractions · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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Simplifying algebraic fractions

Mathematics · Algebra · ages 14-15
Name ______________________   Date ____________

  1. A nonzero factor multiplying the entire numerator and entire denominator may be cancelled, while all original denominator restrictions are retained.

    Circle one:   True   False

  2. For x ≠ -5, (x + 5)/(x + 5)² simplifies to 1.

    Circle one:   True   False

  3. The fraction (x² - 81)/(x + 9) is defined and has value zero at x = 9.

    Circle one:   True   False

  4. What is the first move when looking for shared factors in (x² - 9)/(x + 3)?

    • Take square roots of both expressions
    • Factorise the numerator and denominator
    • Delete matching terms inside each sum
    • Expand every expression before checking

  5. Which inputs are excluded from the original fraction (x - 2)/(x² - 2x)?

    • 0 and -2
    • -2 and 2
    • 0 and 2
    • 1 and 2

  6. Sam deletes the additive x terms from (x + 5)/(x + 2), leaving 5/2. Why is this cancellation not valid?

    • The deleted x terms are not shared whole factors.
    • The fraction has no allowed values of x to check.
    • The denominator becomes zero at every value of x.
    • The numerator must be squared before division is allowed.

  7. Simplify (x squared + x - 6)/(x - 2).

    • x - 3
    • x + 2
    • (x + 3)(x - 2)
    • x + 3

  8. (x squared - 25)/(x + 5) simplifies to x - 5, so x can be -5.

    Circle one:   True   False

  9. Simplify (x cubed - 9x)/(x squared - 9).

    • x
    • x - 3
    • (x - 3)(x + 3)
    • x(x - 3)

  10. A fraction with a polynomial numerator simplifies to x + 5 for x ≠ 1. Its original denominator was x - 1. What was its numerator?

    • x squared - 4x - 5
    • x squared + 4x + 5
    • x squared + 4x - 5
    • x squared + 9x - 5
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Simplifying algebraic fractions W1-mt_scDNSoz3eW-s1

  1. True · Cancellation divides both entire expressions by the same nonzero factor. It does not permit any input excluded by the original denominator.
  2. False · The denominator has two copies of x + 5. Only one cancels, leaving 1/(x + 5), with x ≠ -5.
  3. True · At x = 9, the denominator is 18 and numerator is 0. Zero divided by 18 is defined and equals zero.
  4. Factorise the numerator and denominator · Factorisation reveals whole factors that may be shared. The numerator is (x - 3)(x + 3), so x + 3 is shared.
  5. 0 and 2 · The original denominator is x(x - 2), which is zero at 0 and 2. Both stay excluded after cancellation.
  6. The deleted x terms are not shared whole factors. · Deleting the terms gives 5/2, but it changes the value. At the allowed input x = 1, the original gives 6/3 = 2, not 2.5.
  7. x + 3 · The top is (x + 3)(x - 2). Cancel the shared (x - 2) to get x + 3, with x not equal to 2.
  8. False · The original denominator x + 5 is zero at x = -5, so the original fraction is undefined there, even though x - 5 is fine.
  9. x · The top is x(x - 3)(x + 3) and the bottom is (x - 3)(x + 3), so both brackets cancel and x remains, with x not equal to 3 or -3.
  10. x squared + 4x - 5 · Multiply the simplified expression by the original denominator: (x + 5)(x - 1) = x² + 4x - 5.
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