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Quizzes > Quizzes for Business > Education

Algebra Factoring Skills Test Quiz

Master Essential Factoring Techniques with Guided Practice

Difficulty: Moderate
Questions: 20
Learning OutcomesStudy Material
Colorful paper art displaying Algebra Factoring Skills Test quiz

Ready to sharpen your algebra factoring skills and tackle quadratics with confidence? This Algebra Factoring Skills Test offers 15 multiple-choice challenges designed for high school and college students aiming to master polynomial factoring. Explore similar practice in the Algebra Escape Room Quiz or dive into advanced concepts with the Algebra: Linear Equations and Graphs Quiz . Each question can be freely modified in our editor, making it easy to customize for any classroom or study group. Discover more quizzes and start improving your factoring expertise today.

What is the greatest common factor (GCF) of 12x^3 and 18x^2?
6x^2
12x
3x^3
18x^2
The GCF is the largest common factor: 12 and 18 share 6, and x^3 and x^2 share x^2, giving 6x^2.
Factor out the GCF from 14x^4 - 21x^3.
7x^3(2x - 3)
7x^2(2x^2 - 3x)
x^3(14x - 21)
14x(x^3 - 1.5x^2)
The greatest common factor is 7x^3; dividing each term yields 2x - 3. Pulling out 7x^3 yields 7x^3(2x-3).
Factor completely: 9x^2 - 16.
(3x - 4)(3x + 4)
(9x - 16)(x + 1)
(3x - 8)(3x + 2)
(x - 4)(9x + 4)
This is a difference of squares: 9x^2 is (3x)^2 and 16 is (4)^2. It factors as (3x-4)(3x+4).
Factor the perfect square trinomial: x^2 + 6x + 9.
(x + 3)^2
(x + 1)(x + 9)
(x - 3)^2
(x + 3)(x + 2)
x^2+6x+9 is a perfect square trinomial: (x+3)^2 expands to x^2 +6x +9.
Factor completely: x^2 + 5x + 6.
(x + 2)(x + 3)
(x + 1)(x + 6)
(x + 3)^2
(x + 6)(x - 1)
We seek two numbers that multiply to 6 and add to 5: 2 and 3. So the factorization is (x+2)(x+3).
Factor completely: 4x^2 - 12x + 8.
4(x - 1)(x - 2)
4(x - 2)(x - 2)
(2x - 4)(2x - 2)
4(x - 4)(x + 2)
First factor out 4: 4(x^2 -3x +2), then factor the trinomial into (x-1)(x-2).
Factor by grouping: x^3 + 2x^2 + 3x + 6.
(x + 2)(x^2 + 3)
x(x^2 + 2x + 3) + 6
(x + 3)(x^2 + 2)
(x + 6)(x^2 + 2x + 1)
Group terms: x^2(x+2) +3(x+2) gives a common factor (x+2), yielding (x+2)(x^2+3).
Factor completely: 6x^2 + 11x + 3.
(2x + 3)(3x + 1)
(6x + 1)(x + 3)
(3x + 3)(2x + 1)
(2x - 3)(3x - 1)
6*3=18 and we need sum 11; split as 9 and 2. Grouping yields factors (2x+3)(3x+1).
Factor completely: 25y^2 - 30y + 9.
(5y - 3)^2
(5y + 3)^2
(25y - 9)(y - 1)
(5y - 9)(5y + 1)
This is a perfect square trinomial: the square root of 25y^2 is 5y and of 9 is 3, with a negative middle term.
Factor completely: x^2 - y^2.
(x - y)(x + y)
(x - y)^2
(x + y)^2
(x - y)(y - x)
x^2 - y^2 is a difference of squares, factoring into (x-y)(x+y).
Factor completely: 3x^2 - 12x - 15.
3(x - 5)(x + 1)
(3x - 5)(x - 3)
3(x + 5)(x - 1)
(x - 3)(3x + 5)
Factor out 3 to get 3(x^2 -4x -5), then factor the quadratic as (x-5)(x+1).
Factor by grouping: 2x^3 - 8x^2 + x - 4.
(x - 4)(2x^2 + 1)
2x^2(x - 4) + (x - 4)
(2x + 1)(x^2 - 4)
(2x - 4)(x^2 + 1)
Group into (2x^3 -8x^2) + (x -4), factor each group as 2x^2(x-4) +1(x-4) to get (x-4)(2x^2+1).
Factor completely: 4a^2 - 9b^2.
(2a - 3b)(2a + 3b)
(4a - 3b)(a + 3b)
(2a - 9b)(2a + b)
(a - 3b)(4a + b)
Difference of squares: 4a^2 is (2a)^2 and 9b^2 is (3b)^2, giving (2a-3b)(2a+3b).
Factor completely: 3x^2 + 14x + 8.
(x + 4)(3x + 2)
(3x + 4)(x + 2)
(x + 8)(3x + 1)
(3x + 8)(x + 1)
Multiply to 3*8=24 and sum to 14; decomposition (3x+2)(x+4) yields the correct expansion.
Factor completely: 8x^3 - 27.
(2x - 3)(4x^2 + 6x + 9)
(4x - 3)(2x^2 + 6x + 9)
(2x - 9)(4x^2 + x + 3)
(8x - 3)(x^2 + 6x + 9)
This is a difference of cubes: a^3 - b^3 factors as (a - b)(a^2 + ab + b^2) with a=2x and b=3.
Factor completely: x^4 - y^4.
(x - y)(x + y)(x^2 + y^2)
(x^2 - y^2)^2
(x^2 + y^2 - 2xy)(x^2 + y^2 + 2xy)
(x - y)(x^3 + xy + y^2)
Use difference of squares twice: x^4 - y^4 = (x^2 - y^2)(x^2 + y^2), then factor x^2 - y^2 further.
Factor completely: x^4 - 16.
(x - 2)(x + 2)(x^2 + 4)
(x^2 - 4)^2
(x - 4)(x + 4)(x^2 + 4)
(x - 2)(x + 2)^2
x^4 -16 is a difference of squares: (x^2-4)(x^2+4), and x^2-4 further factors as (x-2)(x+2).
Factor completely: x^3 - 4x^2 - x + 4.
(x - 4)(x - 1)(x + 1)
(x - 2)(x^2 - 2)
(x - 4)(x^2 + 1)
(x + 4)(x^2 - 1)
Group terms: x^2(x -4) -1(x -4) = (x -4)(x^2 -1), and x^2 -1 factors as (x-1)(x+1).
Factor completely: 2x^3 + x^2 - 8x - 4.
(2x + 1)(x - 2)(x + 2)
(2x + 1)(x^2 - 4)
(x + 1)(2x^2 - 8)
(2x - 1)(x^2 + 4)
Group into (2x^3 + x^2) + (-8x -4), factor each: x^2(2x+1) -4(2x+1), yielding (2x+1)(x^2 -4), then further.
Factor completely: a^3 + 8b^3.
(a + 2b)(a^2 - 2ab + 4b^2)
(a + 2b)(a^2 + 2ab + 4b^2)
(a + 4b)(a^2 - 4ab + 16b^2)
(a + 2b)(a^2 - 4ab + 4b^2)
Sum of cubes formula: a^3 + b^3 = (a + b)(a^2 - ab + b^2). Here b=2b so a^3 + (2b)^3.
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Learning Outcomes

  1. Identify greatest common factors in expressions
  2. Apply the distributive property to simplify and factor expressions
  3. Factor trinomials using decomposition techniques
  4. Master factoring patterns like difference of squares
  5. Demonstrate factoring of perfect square trinomials
  6. Evaluate the correctness of factoring solutions

Cheat Sheet

  1. Identify the Greatest Common Factor (GCF) - Gear up for some algebra sleuthing by spotting the biggest factor hiding in every term! Pulling out the GCF, like 3x in 6x² + 9x, rewrites the problem as 3x(2x + 3), making the next steps a breeze. Factor Special Products (Intermediate Algebra)
    2. Factor Special Products (Intermediate Algebra) on OpenStax
  2. Apply the Distributive Property - Think of distribution as spreading peanut butter evenly over bread - each term inside the parentheses gets a share! Expanding 2x(3x + 4) into 6x² + 8x (and vice versa) solidifies your command of both expanding and factoring. Factor Special Products (Elementary Algebra)
    3. Factor Special Products (Elementary Algebra) on OpenStax
  3. Factor Trinomials by Decomposition - Break the middle term into two parts that multiply to the constant and add to the linear coefficient - it's like solving a mini-puzzle! For x² + 5x + 6, splitting 5x into 2x + 3x yields (x + 2)(x + 3) in no time. Factor Special Products (Intermediate Algebra 2e)
    4. Factor Special Products (Intermediate Algebra 2e) on OpenStax
  4. Recognize Perfect Square Trinomials - These trinomials dress up like a square: a² + 2ab + b² pops right into (a + b)². Spotting x² + 6x + 9 as (x + 3)² makes factoring feel like second nature. Perfect Square Trinomials (Intermediate Algebra)
    5. Perfect Square Trinomials (Intermediate Algebra) on OpenStax
  5. Master the Difference of Squares - When you see a² - b², it's a red-carpet invite to (a + b)(a - b)! For example, x² - 16 easily factors to (x + 4)(x - 4). This quick trick unlocks binomials in a snap. Difference of Squares (Elementary Algebra)
    6. Difference of Squares (Elementary Algebra) on OpenStax
  6. Understand Sum & Difference of Cubes - Cubes have their own VIP formulas: a³ + b³ = (a + b)(a² - ab + b²) and a³ - b³ = (a - b)(a² + ab + b²). Factoring x³ - 8 into (x - 2)(x² + 2x + 4) showcases this powerful pattern. Sum/Difference of Cubes (Intermediate Algebra 2e)
    7. Sum/Difference of Cubes (Intermediate Algebra 2e) on OpenStax
  7. Practice Factoring by Grouping - Got four terms? Group them into pairs and factor each pair - like teaming up players for the perfect play. For x³ + 3x² + 2x + 6, grouping yields (x² + 2)(x + 3). High-five your way through complex polynomials! Factoring by Grouping (Intermediate Algebra)
    8. Factoring by Grouping (Intermediate Algebra) on OpenStax
  8. Always Check for a GCF First - Before any fancy method, sweep the room for a GCF to simplify the setup. Pulling out 4x from 4x² + 8x to get 4x(x + 2) makes subsequent techniques smoother. GCF First (Elementary Algebra)
    9. GCF First (Elementary Algebra) on OpenStax
  9. Verify by Expanding - Double-check your factoring by multiplying factors back out - it's your factoring proof of purchase! Expanding (x + 2)(x + 3) to x² + 5x + 6 guarantees you nailed it. Verify by Expanding (Intermediate Algebra 2e)
    10. Verify by Expanding (Intermediate Algebra 2e) on OpenStax
  10. Practice Regularly - Factoring is like learning an instrument: daily riffs build muscle memory. Mix up problems - from GCF to cubes - to become a factoring rockstar in no time. Factor Polynomials (Socratic)
    11. Factor Polynomials (Socratic)
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