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Math 10C

Math 10C · Worksheets

Polynomials and factoring

Ten questions of mixed difficulty, covering Polynomials. Print it, or work through it on screen — the answer key starts on its own page.

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Polynomials and factoring

Math 10C · maddyhelps.com

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Circle the best answer for each question. Show your work in the space provided.

  1. Factored, x² − 25 is

    1. a) x(x − 25)
    2. b) (x − 5)²
    3. c) (x − 25)(x + 1)
    4. d) (x − 5)(x + 5)

  2. Expanded, (x + 3)² is

    1. a) x² + 9
    2. b) x² + 3x + 9
    3. c) x² + 6x + 9
    4. d) x² + 6x + 3

  3. Factored, 4x² − 49 is

    1. a) (2x − 7)²
    2. b) (4x − 7)(x + 7)
    3. c) (x − 7)(4x + 7)
    4. d) (2x − 7)(2x + 7)

  4. Factored, 3x² + 7x + 2 is

    1. a) (3x + 7)(x + 2)
    2. b) 3(x + 1)(x + 2)
    3. c) (3x + 1)(x + 2)
    4. d) (3x + 2)(x + 1)

  5. Expanded, (x + 2)(x + 5) is

    1. a) x² + 10
    2. b) x² + 10x + 7
    3. c) x² + 7x + 10
    4. d) x² + 7x + 7

  6. Factored completely, 2x² + 10x + 12 is

    1. a) 2(x + 1)(x + 6)
    2. b) (2x + 4)(x + 3)
    3. c) 2(x + 2)(x + 3)
    4. d) (x + 2)(x + 6)

  7. Expanded, 3(x + 4) is

    1. a) 3x·4
    2. b) 3x + 12
    3. c) 3x + 4
    4. d) x + 12

  8. Factored, x² − 3x − 10 is

    1. a) (x − 5)(x − 2)
    2. b) (x + 5)(x − 2)
    3. c) (x − 10)(x + 1)
    4. d) (x − 5)(x + 2)

  9. Factored, x² + 9x + 20 is

    1. a) (x − 4)(x − 5)
    2. b) (x + 4)(x + 5)
    3. c) (x + 2)(x + 10)
    4. d) (x + 1)(x + 20)

  10. Expanded, (2x − 3)(x + 4) is

    1. a) 2x² − 12
    2. b) 2x² + 5x + 12
    3. c) 2x² + 11x − 12
    4. d) 2x² + 5x − 12

Answer key · Polynomials and factoring

Math 10C · maddyhelps.com

  1. d) (x − 5)(x + 5) — A difference of squares always splits into a difference times a sum. Note that a <i>sum</i> of squares like x² + 25 does not factor over the real numbers at all.
  2. c) x² + 6x + 9 — A square means the bracket times itself, so the middle term is twice the product: x² + 3x + 3x + 9. Writing x² + 9 is the single most common algebra error at this level.
  3. d) (2x − 7)(2x + 7) — This is still a difference of squares, with a = 2x because (2x)² = 4x², and b = 7. Recognising that the first term is a perfect square even with a coefficient is the step being tested.
  4. c) (3x + 1)(x + 2) — Check by expanding: (3x + 1)(x + 2) gives 3x² + 6x + x + 2 = 3x² + 7x + 2 ✓. The near-miss (3x + 2)(x + 1) gives a middle term of 5x, so which bracket gets the 2 matters.
  5. c) x² + 7x + 10 — Every term in the first bracket multiplies every term in the second: x² + 5x + 2x + 10, then collect to x² + 7x + 10. Checking with x = 1 gives 3 × 6 = 18 on the left and 1 + 7 + 10 = 18 on the right ✓.
  6. c) 2(x + 2)(x + 3) — Take out the common factor 2 first, leaving x² + 5x + 6 = (x + 2)(x + 3). The option (2x + 4)(x + 3) expands correctly but is not <i>completely</i> factored, because 2x + 4 still has a common factor.
  7. b) 3x + 12 — Distribute the 3 over both terms in the bracket. Multiplying only the first term is the error that produces 3x + 4, and it is worth checking every expansion by substituting a number.
  8. d) (x − 5)(x + 2) — Two numbers multiplying to −10 and adding to −3: that is −5 and +2. A negative constant term means the signs in the brackets are different, and the bigger number takes the sign of the middle term.
  9. b) (x + 4)(x + 5) — Find two numbers multiplying to 20 and adding to 9: 4 and 5. The pairs 2 and 10 or 1 and 20 both multiply to 20 but add to 12 and 21, so they fail the second test.
  10. d) 2x² + 5x − 12 — 2x² + 8x − 3x − 12, which collects to 2x² + 5x − 12. Substituting x = 1 gives (−1)(5) = −5 on the left and 2 + 5 − 12 = −5 on the right ✓.