Factoring Techniques Quiz

Grade 10 · Math · Quiz

Test your factoring skills across GCF, trinomials, difference of squares, and perfect squares.

Questions

  1. Factor: 4x² - 25

    1. (2x - 5)²
    2. (4x - 25)(x + 1)
    3. (2x + 5)(2x - 5)
    4. (2x - 5)(2x - 5)
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    Answer: (2x + 5)(2x - 5)

    4x²-25 = (2x)²-5². Difference of squares: (2x+5)(2x-5).

  2. GCF of 12x⁴ - 8x³ + 4x² is ___.

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    Answer: 4x²

    Coefficients GCF: GCF(12,8,4)=4. Variable: lowest power is x². So GCF = 4x².

  3. Factor: x² - x - 6

    1. (x - 3)(x + 2)
    2. (x + 3)(x - 2)
    3. (x - 6)(x + 1)
    4. (x + 6)(x - 1)
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    Answer: (x - 3)(x + 2)

    Need factors of -6 that add to -1: -3 and 2. Check: -3×2=-6 ✓, -3+2=-1 ✓. Factor: (x-3)(x+2).

  4. 9x² + 6x + 1 factors as (3x + 1)².

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    Answer: True

    9x²+6x+1 = (3x)²+2(3x)(1)+1² = (3x+1)². This matches the perfect square trinomial pattern.

  5. Factor by grouping: 2x³ + 4x² + 3x + 6

    1. (x + 2)(2x² + 3)
    2. (2x + 3)(x² + 2)
    3. (x + 3)(2x² + 2)
    4. (2x² - 3)(x - 2)
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    Answer: (x + 2)(2x² + 3)

    Group: (2x³+4x²)+(3x+6) = 2x²(x+2)+3(x+2) = (x+2)(2x²+3).

  6. The factored form of a² - b² is (a + b)(a - ___).

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    Answer: b

    The difference of squares formula: a²-b² = (a+b)(a-b). The second factor has (a-b).

  7. Which polynomial is prime (cannot be factored over the integers)?

    1. x² - 4
    2. x² + 4
    3. x² - 9
    4. x² - 2x - 3
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    Answer: x² + 4

    x²+4 is a sum of squares — it doesn't factor over the reals. The others all factor: differences of squares or factorable trinomials.

  8. x² - 14x + 49 = (x - 7)².

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    Answer: True

    x²-14x+49 = x²-2(7)x+7² = (x-7)². The middle term is -2ab with b=7.

  9. Factor completely: 3x² - 12

    1. 3(x² - 4)
    2. 3(x - 2)(x + 2)
    3. (3x - 6)(x + 2)
    4. 3(x - 4)(x + 1)
    Show answer

    Answer: 3(x - 2)(x + 2)

    Factor GCF first: 3(x²-4). Then factor the difference of squares: 3(x-2)(x+2). Complete factoring.

  10. To check a factored answer, you should ___ the factors together.

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    Answer: multiply

    Multiplying factors back out (FOIL or distribution) verifies the factoring is correct.

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