Polynomials and Factoring Quiz

Grade 9 · Math · Quiz

Test your skills on polynomial operations, identifying special forms, and applying multiple factoring techniques.

Questions

  1. What is the degree of the polynomial 5x⁴ − 3x² + 7x − 2?

    1. 2
    2. 3
    3. 4
    4. 7
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    Answer: 4

    The degree is determined by the highest exponent. The term 5x⁴ has the highest exponent of 4, so the degree is 4.

  2. Factor completely: 3x² − 75

    1. 3(x − 5)(x + 5)
    2. 3(x² − 25)
    3. (3x − 15)(x + 5)
    4. (x − 5)(3x + 15)
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    Answer: 3(x − 5)(x + 5)

    First factor out the GCF of 3: 3(x² − 25). Then apply difference of squares: 3(x − 5)(x + 5).

  3. Which expression is equivalent to (x + 4)(x − 4)?

    1. x² + 16
    2. x² − 16
    3. x² + 8x − 16
    4. x² − 8x + 16
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    Answer: x² − 16

    (x + 4)(x − 4) is a difference of squares pattern: x² − 4² = x² − 16.

  4. Factor x² − 3x − 18 completely.

    1. (x − 6)(x + 3)
    2. (x + 6)(x − 3)
    3. (x − 9)(x + 2)
    4. (x + 9)(x − 2)
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    Answer: (x − 6)(x + 3)

    Find two numbers that multiply to −18 and add to −3: −6 and +3. So x² − 3x − 18 = (x − 6)(x + 3).

  5. Factor 6x² − 7x − 3 completely.

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

    AC method: 6 × (−3) = −18. Factors of −18 that add to −7: −9 and +2. Rewrite: 6x² − 9x + 2x − 3 = 3x(2x − 3) + 1(2x − 3) = (3x + 1)(2x − 3).

  6. Which of the following is a perfect square trinomial?

    1. x² + 6x + 8
    2. x² + 4x + 4
    3. x² − 5x + 6
    4. x² + 3x + 9
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    Answer: x² + 4x + 4

    x² + 4x + 4 = (x + 2)² because 2² = 4 and 2(1)(2) = 4, matching a² + 2ab + b² = (a + b)².

  7. 4x² − 12x + 9 factors as (2x − 3)².

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    Answer: True

    Check: (2x − 3)² = 4x² − 12x + 9. Indeed 2·(2x)·(−3) = −12x and (−3)² = 9. Confirmed.

  8. Factor x³ + x² − 6x by first factoring out the GCF. The result is x(x + 3)(x − __).

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    Answer: 2

    Factor GCF x first: x(x² + x − 6). Then factor the trinomial: find numbers that multiply to −6 and add to 1, which are +3 and −2. Result: x(x + 3)(x − 2).

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