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Quizzes > Mathematics

Factoring Quiz: Practice Polynomials and Check Your Work

Short, self-graded factoring polynomials quiz with instant feedback.

Editorial: Review CompletedCreated By: Stefano PiccoloUpdated Aug 27, 2025
Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
paper art illustration with numbers variables and factorization symbols on sky blue background for a factoring quiz

This factoring polynomials quiz helps you practice turning expressions into products and check your steps in minutes. If you want a quick tune-up first, review the distributive property quiz or sharpen basics with the combine like terms quiz. Afterward, broaden your skills with an algebra practice quiz.

Factor completely over the integers: x^2 - 9
(x - 9)
(x - 3)(x + 3)
Prime
(x - 9)(x + 1)
undefined
Factor the trinomial: x^2 + 7x + 12
(x + 2)(x + 5)
(x + 12)(x - 1)
(x + 3)(x + 4)
(x + 6)(x + 1)
undefined
Factor out the greatest common factor: 6x^3y - 9x^2y^2
6xy(x^2 - 3y^2)
3xy(2x^3 - 3x^2y)
3x^2y(2x - 3y)
x^2y(6x - 9y^2)
undefined
Factor completely: 4y^2 - 25
(y - 5)(4y + 5)
(4y - 25)(y + 1)
(2y - 5)(2y + 5)
Prime
undefined
Factor the trinomial: x^2 - x - 12
(x - 3)(x + 4)
(x - 4)(x + 3)
(x - 2)(x + 6)
(x - 6)(x + 2)
undefined
Factor by grouping: x^3 + 3x^2 + 2x + 6. Which option lists the linear factor first?
(x^2 + 2)(x + 3)
(x + 1)(x^2 + 2x + 6)
(x + 2)(x^2 + 3)
(x + 3)(x^2 + 2)
undefined
Factor the difference of cubes: x^3 - 8
(x - 2)(x^2 + 2x + 4)
(x + 2)(x^2 - 2x + 4)
(x - 2)(x^2 - 2x + 4)
(x + 2)(x^2 + 2x + 4)
undefined
Factor the sum of cubes: 27y^3 + 1
(3y - 1)(9y^2 + 3y + 1)
(3y + 1)(9y^2 + 3y + 1)
(3y - 1)(9y^2 - 3y + 1)
(3y + 1)(9y^2 - 3y + 1)
undefined
Factor completely: x^4 - 16
(x^2 - 16)(x^2 + 1)
(x - 4)(x + 4)
(x - 2)(x + 2)(x^2 + 4)
(x^2 - 4)(x^2 + 4)
undefined
Factor the quadratic: 3x^2 - 11x - 4
(3x + 1)(x - 4)
(3x - 4)(x + 1)
(x - 1)(3x - 4)
(3x - 1)(x + 4)
undefined
Factor by grouping: 2ab + 8a + 3b + 12
(2a + 8)(b + 3)
(b + 4)(2a + 3)
(2a + 3)(b + 4)
(a + 4)(2b + 3)
undefined
Factor completely: 5x^3 - 20x
5x(x^2 - 4)
x(5x^2 - 20)
5(x^3 - 4x)
5x(x - 2)(x + 2)
undefined
Factor the quartic: x^4 - 5x^2 + 4 into quadratic factors over the integers.
(x^2 - 1)(x^2 - 4)
Prime
(x^2 - 2)(x^2 - 3)
(x - 1)(x - 4)(x + 1)(x + 4)
undefined
Factor completely over integers: 2x^2 + x + 5
Prime
(2x - 5)(x - 1)
(2x + 5)(x + 1)
(x + 5)(2x + 1)
undefined
Factor completely: 3x^3 - 3x^2 - 12x + 12
3(x - 1)(x^2 - 4)
3(x^3 - x^2 - 4x + 4)
3(x - 1)(x - 2)(x + 2)
3x(x^2 - x - 4)
undefined
Factor the trinomial: 10y^2 - 13y - 3
(10y + 1)(y - 3)
(5y - 3)(2y + 1)
(5y - 1)(2y + 3)
(5y + 1)(2y - 3)
undefined
Factor completely over integers: x^2 - 2x + 2
(x - 2)(x + 1)
(x - 1)^2 + 1
Prime
(x - 1)(x - 2)
undefined
Factor completely: x^6 - 64
(x^2 - 4)(x^4 + 16)
(x - 2)(x + 2)(x^2 - 2x + 4)(x^2 + 2x + 4)
(x^2 - 8)(x^4 + 8)
(x^3 - 8)(x^3 + 8)
undefined
Factor completely: 9x^4 - 25y^4
(3x - 5y)(3x + 5y)(3x^2 + 5y^2)
(3x^2 - 5y^2)(3x^2 + 5y^2)
(3x - 5y)(3x + 5y)(3x^2 + 5y^2) is incorrect; correct is (3x^2 - 5y^2)(3x^2 + 5y^2) = (3x - 5y)(3x + 5y)(3x^2 + 5y^2)
(3x - 5y)(3x + 5y)(x^2 + 25y^2/9)
undefined
Factor completely: 2x^4 + 3x^2 - 2
(x^2 + 2)(2x^2 - 1)
Prime
(2x^2 + 1)(x^2 - 2)
(2x^2 - 1)(x^2 + 2) is incorrect; correct is (2x^2 + 1)(x^2 - 2)
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Study Outcomes

  1. Understand Fundamental Factoring Concepts -

    Gain clarity on core principles of factoring simple polynomials and extracting greatest common factors.

  2. Identify Trinomial Patterns -

    Recognize common structures in quadratics and trinomials to streamline your polynomial factoring practice.

  3. Apply Systematic Factoring Methods -

    Use step-by-step strategies to tackle both simple and difficult factoring questions with confidence.

  4. Analyze Instant Quiz Feedback -

    Interpret real-time results from the factoring polynomials quiz to pinpoint strengths and areas for improvement.

  5. Evaluate Factorization Accuracy -

    Assess your solutions for correctness, reinforcing algebra foundations and reducing errors.

  6. Enhance Speed and Precision -

    Boost your proficiency in factor polynomials exercises through repeated practice and challenge yourself effectively.

Cheat Sheet

  1. Greatest Common Factor (GCF) -

    Always start by pulling out the greatest common factor to simplify polynomials into manageable parts; for example, 6x^3 + 9x^2 becomes 3x^2(2x + 3). Recognizing the GCF first is endorsed by MIT OpenCourseWare to reduce errors in subsequent steps. This foundational move saves time when tackling both simple factoring questions and more difficult factoring questions.

  2. Factoring Simple Trinomials -

    For expressions like x^2 + 5x + 6, look for two numbers that multiply to 6 and add to 5 - in this case, 2 and 3 - so x^2 + 5x + 6 = (x + 2)(x + 3). This "product-sum" method is highlighted by Khan Academy as a reliable strategy for polynomial factoring practice. A handy mnemonic is "Find two numbers that sum high and multiply wide."

  3. Difference of Squares -

    Recognize patterns of the form a^2 − b^2, which factor as (a − b)(a + b); for example, x^2 − 9 = (x − 3)(x + 3). University of Cambridge notes this as one of the simplest yet most powerful factoring techniques. Spotting this pattern quickly boosts confidence on both simple and challenging factoring polynomials quizzes.

  4. Perfect Square Trinomials -

    Identify expressions matching a^2 ± 2ab + b^2, which factor to (a ± b)^2, such as x^2 + 6x + 9 = (x + 3)^2. According to Purplemath, recognizing these can cut factoring time in half. A quick check: does the first and last term share a perfect-square relationship?

  5. Factoring by Grouping -

    When dealing with four-term polynomials like x^3 + 3x^2 + 2x + 6, group to factor: (x^3 + 3x^2) + (2x + 6) = x^2(x + 3) + 2(x + 3) = (x + 3)(x^2 + 2). This method, supported by Wolfram MathWorld, is essential for complex factor polynomials exercises. Always look for a common binomial in grouped terms to simplify efficiently.

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