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Quizzes > Mathematics > Algebra

Factoring Polynomials Practice Quiz: Are You Ready to Ace It?

Tackle factorization of polynomials practice problems and conquer polynomial factoring practice!

Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
Paper art illustration promoting a free factoring polynomials quiz on a sky blue background

Ready to master factoring polynomials practice? Dive into our Free Factoring Polynomials Practice Quiz: Test Your Skills and sharpen your algebra edge with engaging factorization of polynomials practice problems. Whether you're brushing up on simple factoring or craving a challenge, these practice problems for factoring polynomials will boost your confidence and build your skills. Check out some simple factoring questions to warm up, then conquer the full factoring quiz . Start now, embrace the challenge, and see how far your polynomial factoring practice can take you!

Factor the expression x² - 9.
(x + 3)²
(x - 9)(x + 1)
(x - 3)(x + 3)
(x - 1)(x + 9)
The expression x² - 9 is a difference of squares and factors into (x - 3)(x + 3). The formula a² - b² = (a - b)(a + b) applies here with a = x and b = 3. This is a fundamental factoring pattern in algebra. For more detailed examples, visit Khan Academy.
Factor the trinomial x² + 5x + 6.
(x + 2)(x + 3)
(x + 1)(x + 6)
(x - 2)(x - 3)
(x + 3)²
We look for two numbers that multiply to 6 and add to 5, which are 2 and 3. Thus x² + 5x + 6 = (x + 2)(x + 3). This is the standard method for factoring simple quadratics. Read more at Khan Academy.
Factor the expression x² - 5x.
(x - 1)(x - 4)
(x - 5)²
(x + 5)(x - 1)
x(x - 5)
Both terms share a common factor x, so we factor it out: x² - 5x = x(x - 5). This is a straightforward use of the distributive property in reverse. See similar examples on Khan Academy.
Factor the expression 4x² - 12x.
(2x - 6)²
4(x - 9)x
2x(2x + 6)
4x(x - 3)
The greatest common factor of 4x² and -12x is 4x, so 4x² - 12x = 4x(x - 3). Factoring out the GCF is the first step in simplifying polynomials. For more on GCF factoring, visit Khan Academy.
Factor the quadratic x² + 2x - 8.
(x + 4)(x - 2)
(x - 8)(x + 1)
(x + 8)(x - 1)
(x - 4)(x + 2)
We need two numbers that multiply to -8 and add to 2; those numbers are 4 and -2. Thus x² + 2x - 8 = (x + 4)(x - 2). This is a common quadratic factoring technique. More practice is available at Khan Academy.
Factor 6x² + x - 2.
(6x - 1)(x + 2)
(3x - 2)(2x + 1)
(2x - 1)(3x + 2)
(6x + 2)(x - 1)
We look for a pair that multiplies to (6)(-2) = -12 and adds to 1; those are 4 and -3. We split the middle term: 6x² + 4x - 3x - 2, then factor by grouping: 2x(3x + 2) -1(3x + 2) = (2x - 1)(3x + 2). For detailed steps, see Khan Academy.
Factor x² - x - 12.
(x - 2)(x + 6)
(x - 6)(x + 2)
(x - 4)(x + 3)
(x + 4)(x - 3)
We need two numbers multiplying to -12 and adding to -1; those are -4 and 3. Thus x² - x - 12 = (x - 4)(x + 3). This method of 'find two numbers' is standard for quadratics. Additional examples can be found at Khan Academy.
Factor 9x² - 25.
(9x - 25)(x + 1)
(9x - 5)(x + 5)
(3x - 5)(3x + 5)
(3x - 25)(3x + 1)
This is a difference of squares: 9x² - 25 = (3x)² - 5², which factors as (3x - 5)(3x + 5). Recognizing the pattern a² - b² makes factoring quick. Learn more about this pattern at Khan Academy.
Factor the sum of cubes x³ - 27.
(x - 3)(x² + 3x + 9)
(x³ - 3)(x + 9)
(x - 9)(x² + 9x + 27)
(x + 3)(x² - 3x + 9)
x³ - 3³ is a difference of cubes, factoring as (x - 3)(x² + 3x + 9) using the formula a³ - b³ = (a - b)(a² + ab + b²). This is key for cubic expressions. See more at Khan Academy.
Factor the expression 8x³ + 27.
(8x + 27)(x² + 1)
(2x + 3)(4x² - 6x + 9)
(4x + 3)(2x² - 3x + 9)
(2x - 3)(4x² + 6x + 9)
8x³ + 27 = (2x)³ + 3³, a sum of cubes. It factors as (2x + 3)(4x² - 6x + 9) by the formula a³ + b³ = (a + b)(a² - ab + b²). For derivation, visit Khan Academy.
Factor x? - 16 completely.
(x - 2)(x + 2)(x² + 4)
(x - 4)(x + 4)(x² + 4)
(x² - 4)(x² - 4)
(x² - 16)(x² + 16)
x? - 16 is a difference of squares twice: x? - 16 = (x² - 4)(x² + 4), then x² - 4 = (x - 2)(x + 2). Combining gives (x - 2)(x + 2)(x² + 4). See a step-by-step on Khan Academy.
Factor by grouping: x³ + x² - x - 1.
(x + 1)²(x - 1)
(x - 1)(x² + x + 1)
(x + 1)(x - 1)²
(x + 1)(x² - x - 1)
Group as (x³ + x²) + ( - x - 1) = x²(x + 1) - 1(x + 1) = (x + 1)(x² - 1) = (x + 1)(x + 1)(x - 1) = (x + 1)²(x - 1). Grouping reveals a common binomial factor. More examples at Khan Academy.
Factor x? + 4 completely over the reals.
(x + 2i)(x - 2i)(x² + 1)
(x² - 2x + 2)²
(x² + 4)(x² + 1)
(x² + 2x + 2)(x² - 2x + 2)
x? + 4 can be rewritten using Sophie Germain's identity: a? + 4b? = (a² + 2ab + 2b²)(a² - 2ab + 2b²) with a=x, b=1. Thus it factors as (x² + 2x + 2)(x² - 2x + 2). For a deeper dive, see Sophie Germain identity on Wikipedia.
Factor completely: 27x³ - 8y³.
(3x + 2y)(9x² - 6xy + 4y²)
(3x - 2y)(9x² + 6xy + 4y²)
(3x - 2y)³
(27x - 8y)(x² + xy + y²)
This is a difference of cubes: (3x)³ - (2y)³, which factors as (3x - 2y)(9x² + 6xy + 4y²) by the formula a³ - b³ = (a - b)(a² + ab + b²). This expands into the given quadratic. For more, visit Khan Academy.
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Study Outcomes

  1. Identify Greatest Common Factors -

    Apply factoring polynomials practice to locate and extract the greatest common factor from various polynomial expressions.

  2. Factor Quadratic Trinomials -

    Use systematic methods to factor trinomials of the form ax² + bx + c into binomial products with confidence.

  3. Recognize Special Products -

    Decompose perfect square trinomials and differences of squares through dedicated factorization of polynomials practice problems.

  4. Factor Complex Polynomials -

    Tackle advanced practice problems for factoring polynomials involving higher-degree terms and multiple variables.

  5. Analyze Factoring Strategies -

    Evaluate and compare different techniques to determine the most efficient approach for each factoring challenge.

  6. Boost Factoring Fluency -

    Practice polynomial factoring exercises to improve speed and accuracy, building algebraic confidence.

Cheat Sheet

  1. Extract the Greatest Common Factor (GCF) -

    Always begin factoring polynomials practice by identifying the GCF in each term. For example, 6x²+9x factors to 3x(2x+3), which simplifies subsequent steps (source: MIT OpenCourseWare). This foundational move boosts confidence and streamlines more complex factorization of polynomials practice problems.

  2. Factor Simple Trinomials (ax²+bx+c) -

    Use the "product-sum" method for quadratics: find two numbers that multiply to a⋅c and add to b. For instance, x²+5x+6 becomes (x+2)(x+3), a key skill in polynomial factoring practice (source: Khan Academy). Mnemonic: "Find the pair that sums and shares" - it really sticks!

  3. Recognize Special Products -

    Memorize patterns like Difference of Squares (a²−b²=(a−b)(a+b)) and Perfect Square Trinomials (a²±2ab+b²=(a±b)²). Spotting these instantly on practice problems for factoring polynomials saves time and power (source: Purplemath). Treat these like algebra's "power moves" for quick wins.

  4. Group for Four-Term Polynomials -

    When faced with ax³+bx²+cx+d, split into two pairs and factor each. For example, x³+3x²+2x+6 = x²(x+3)+2(x+3) = (x+3)(x²+2) (source: University of Illinois). This grouping trick unlocks tougher factorization of polynomials practice problems.

  5. Verify with FOIL and Practice -

    Always multiply your factors back (First, Outside, Inside, Last) to ensure accuracy. Regularly tackle timed polynomial factoring practice sets from reputable sites like the American Mathematical Society to build speed and precision. Consistent practice polishes skills and builds confidence for any quiz or exam.

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