Polynomial Division and the Factor Theorem · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Polynomial Division and the Factor Theorem

Mathematics · Algebra · ages 16-17
Name ______________________   Date ____________

  1. If f(3) = 5, then (x - 3) is a factor of f(x).

    Circle one:   True   False

  2. What is the remainder when x³ + 2x - 5 is divided by (x - 1)?

    Answer: ______________

  3. x³ + 5x² - 2x + 8 is divided by (x + 1). What is the first term of the quotient?

    • x³
    • x
    • x²
    • 1

  4. f(x) = x³ - x² - 8x + 12. Work out f(2).

    Answer: ______________

  5. Divide x³ + 3x² - 4x - 12 by (x + 3). What is the quotient?

    • x² + 3x - 4
    • x² + 4
    • x² - 4x
    • x² - 4

  6. Leah divides x³ + 2x² - 5x - 6 by (x + 1) and gets a quotient of x² + 3x - 6, but her check fails. What is the correct quotient?

    • x² + 3x - 6
    • x² + x - 6
    • x² - x - 6
    • x² + x + 6

  7. When f(x) is divided by (x - 5) the remainder is 12. What is f(5)?

    • 0
    • 12
    • 5
    • -12

  8. Why does factorising a cubic usually start with a trial substitution?

    • a cubic can never break into three brackets
    • the pair method works on cubics as well as quadratics
    • you have to complete the square on the cubic first
    • you need one root before the pair method works

  9. x³ - x² - 4x + 4 factorises as (x - 1)(x - 2)(x + 2).

    Circle one:   True   False

  10. Which of these is a factor of f(x) = x³ - 2x² - 11x + 12?

    • (x - 3)
    • (x + 1)
    • (x + 4)
    • (x + 3)
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Answer key

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Polynomial Division and the Factor Theorem W1-mt_haL2UkGa5C-s1

  1. False · f(3) is the remainder on dividing by (x - 3), and here it is 5, not 0. A factor has to leave nothing over.
  2. -2 · The remainder theorem says the remainder is f(1) = 1 + 2 - 5 = -2. No long division is needed.
  3. x² · The first step divides the leading term of the polynomial by the leading term of the divisor: x³ divided by x is x².
  4. 0 · f(2) = 8 - 4 - 16 + 12. That is 8 - 4 = 4, then 4 - 16 = -12, then -12 + 12 = 0.
  5. x² - 4 · x² times (x + 3) is x³ + 3x², which clears the first two terms. Then -4 times (x + 3) is -4x - 12, which clears the rest. The quotient is x² - 4.
  6. x² + x - 6 · After x²(x + 1) = x³ + x², what is left is x² - 5x - 6, and x² divided by x is x, not 3x. The quotient is x² + x - 6.
  7. 12 · The remainder theorem works both ways: the remainder on dividing by (x - 5) is f(5), so f(5) = 12.
  8. you need one root before the pair method works · There is no pair of numbers to spot in a cubic. Finding one root gives a factor, and dividing it out leaves the quadratic the pair method can handle.
  9. True · (x - 2)(x + 2) is x² - 4, and (x - 1)(x² - 4) expands to x³ - x² - 4x + 4, so the three brackets are right.
  10. (x + 3) · Testing the root each bracket names: f(3) = -12, f(-1) = 20, f(-4) = -40, and f(-3) = -27 - 18 + 33 + 12 = 0. Only -3 works.
Worksheet · LightMySky