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Quizzes > High School Quizzes > Mathematics

Ultimate Factoring Practice Quiz

Boost your skills with grouping practice problems

Difficulty: Moderate
Grade: Grade 8
Study OutcomesCheat Sheet
Colorful paper art promoting Factor Frenzy Grouping Edition algebra quiz for high school students.

Factor the expression: ab + ac + db + dc.
(a+d)(b+c)
a(b+d) + c(b+d)
(a+b)(c+d)
(a+c)(b+d)
Group the terms as (ab + ac) and (db + dc). Factoring out a from the first group and d from the second reveals the common binomial (b+c), resulting in (a+d)(b+c).
Factor the expression: 4x + 4y + 7x + 7y.
(x+7)(x+4)
(x+y)(7x+4y)
(4x+7y)(x+y)
11(x+y)
First, group the terms into (4x+4y) and (7x+7y). Factoring out the common binomial (x+y) from both groups gives 4(x+y) + 7(x+y), which can be combined as 11(x+y).
Factor the expression: xz + xw + yz + yw.
(xz+yz)+(xw+yw)
(x+w)(y+z)
(x+y)(z+w)
(x+z)(y+w)
Group the expression as (xz + xw) and (yz + yw). Factoring x from the first group and y from the second gives x(z+w) + y(z+w), which factors to (x+y)(z+w).
Factor the expression: 5a + 5b + 2a + 2b.
5a+2b
(a+b)(5a+2b)
(a+2)(b+5)
7(a+b)
Group the terms as (5a + 5b) and (2a + 2b). Factoring 5 from the first group and 2 from the second yields 5(a+b) + 2(a+b). The common binomial (a+b) factors out, giving 7(a+b).
Factor the expression: 6p + 9q + 2p + 3q.
2(3p+3q)
4(2p+3q)
(2p+3q)(6p+9q)
3(2p+3q)
Regroup the expression as (6p+9q) and (2p+3q). Factoring 3 from the first group and 1 from the second gives 3(2p+3q) + 1(2p+3q). Factoring the common binomial (2p+3q) results in 4(2p+3q).
Factor the expression: 3x² - 9xy + 2x - 6y.
(3x-2)(x-3y)
(x-3y)(3x+2)
(3x-9y)(x+2)
(3x+2)(x+3y)
Group the terms as (3x² - 9xy) and (2x - 6y). Factoring 3x from the first and 2 from the second yields 3x(x-3y) + 2(x-3y). The common factor (x-3y) can then be factored out to obtain (x-3y)(3x+2).
Factor the expression: 6x + 18 + x + 3.
(x+3)
(x+18)(x+3)
7(x+3)
6(x+3)
Notice that both groups, (6x+18) and (x+3), have the common binomial (x+3). Factoring it out combines the coefficients to yield 7(x+3).
Factor the expression: 2a² + 8a + 3a + 12.
2a(a+4)+3(a+4)
(a+4)(2a+3)
(a+2)(4a+6)
(2a+8)(a+12)
Group the expression into (2a²+8a) and (3a+12). Factoring 2a from the first group and 3 from the second gives 2a(a+4) + 3(a+4). The common factor (a+4) is then factored out, yielding (a+4)(2a+3).
Factor the expression: 4x² + 6xy + 2x + 3y.
(4x+2)(x+3y)
(2x+3y)(2x+1)
(2x+3)(x+1)
(2x+3y)(2x-1)
Divide the polynomial into groups: (4x²+6xy) and (2x+3y). Factoring 2x from the first group and 1 from the second reveals the common factor (2x+3y). This results in the product (2x+3y)(2x+1).
Factor the expression: 5m² + 15mn + 2m + 6n.
(5m+15n)(m+2)
(m+5)(3n+2m)
5m²+15mn+2m+6n
(m+3n)(5m+2)
Group the expression as (5m²+15mn) and (2m+6n). Factoring 5m from the first group and 2 from the second gives 5m(m+3n) + 2(m+3n). Factoring out the common binomial (m+3n) leads to (m+3n)(5m+2).
Factor the expression: 12a + 18b + 4a + 6b.
8(2a+3b)
(2a+3b)(12a+18b)
6(2a+3b)
2(6a+9b)
Group the terms as (12a+18b) and (4a+6b). Factoring 6 from the first group and 2 from the second yields 6(2a+3b) + 2(2a+3b). Factoring out the common binomial (2a+3b) results in 8(2a+3b).
Factor the expression: 9x² + 12xy + 3x + 4y.
3x(3x+4y)+ (3x+4y)
(3x+4y)(3x+1)
(9x+3)(4y+1)
(3x+4y)(x+3)
Group the polynomial as (9x²+12xy) and (3x+4y). Factoring 3x from the first group and 1 from the second gives 3x(3x+4y) + 1(3x+4y). Factoring out the common binomial (3x+4y) results in (3x+4y)(3x+1).
Factor the expression: x³ + 3x² + x + 3.
(x+1)(x²+3)
(x+3)(x²+1)
x²(x+3) + (x+3)
(x+3)(x+1)
Group the expression as (x³+3x²) and (x+3). Factoring x² from the first group yields x²(x+3), and the second group is 1*(x+3). Factoring out the common binomial (x+3) gives (x+3)(x²+1).
Factor the expression: 6x² + 2xy + 9x + 3y.
(2x+3)(3x-y)
(3x+y)(2x+3)
(6x²+2xy)+(9x+3y)
(3x+y)(2x-3)
Separate the expression into (6x²+2xy) and (9x+3y). Factoring 2x from the first group and 3 from the second produces 2x(3x+y) + 3(3x+y). The common factor (3x+y) is then factored out, resulting in (3x+y)(2x+3).
Factor the expression: 6m² + 10mn + 3m + 5n.
(2m+3m)(5n+5n)
(3m+10n)(m+1)
2m(3m+5n)+3m+5n
(3m+5n)(2m+1)
Group the terms as (6m²+10mn) and (3m+5n). Factoring 2m from the first group and 1 from the second gives 2m(3m+5n) + 1(3m+5n). Factoring out the common binomial (3m+5n) yields (3m+5n)(2m+1).
Factor the expression: 2xy + 4y² + 3x + 6y.
2y(x+2y)+3(x+2y)
(2xy+3x)(4y+6y)
(x+2y)(2y-3)
(x+2y)(2y+3)
Group the expression as (2xy+4y²) and (3x+6y). Factoring 2y from the first group and 3 from the second yields 2y(x+2y) + 3(x+2y). The common binomial (x+2y) factors out, resulting in (x+2y)(2y+3).
Factor completely the expression: 4x³ + 8x² + 5x + 10.
(x+2)(4x²+5)
(x+2)(4x²-5)
(4x+5)(x²+2)
4x²(x+2)+5(x+2)
Group the first two terms and the last two terms: 4x³+8x² and 5x+10. Factoring 4x² from the first group and 5 from the second reveals the common factor (x+2), which gives the product (x+2)(4x²+5).
Factor the expression: 3a²b + 9ab² + 2ab + 6b².
(a+3b)(3a+2b)
b(a+3b)(3a+2)
(a+3b)(3ab+2)
b(3a+2)(a+9b)
Group the terms into (3a²b+9ab²) and (2ab+6b²). Factoring 3ab and 2b respectively gives 3ab(a+3b) + 2b(a+3b). Factoring the common binomial (a+3b) and then extracting b results in b(a+3b)(3a+2).
Factor the expression: 5x³ + 10x²y + x + 2y.
(x+2y)(5x²-1)
(x+2y)(5x²+1)
(5x+1)(x²+2y)
5x²(x+2y)+(x+2y)
Group the terms as (5x³+10x²y) and (x+2y). Factoring out 5x² from the first group and 1 from the second group highlights the common binomial (x+2y), resulting in (x+2y)(5x²+1).
Factor completely the expression: 6x²y + 9xy² + 4x² + 6xy.
x(2x+3y)(3y+2)
(2x+3y)(3xy+2)
x(2x+3y)(3y-2)
(2x+3y)(x+3y+2)
Separate the expression into two groups: (6x²y+9xy²) and (4x²+6xy). Factoring 3xy from the first and 2x from the second yields 3xy(2x+3y) + 2x(2x+3y). Factoring out the common binomial (2x+3y) leads to (2x+3y)(3xy+2x), which can be written as x(2x+3y)(3y+2).
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Study Outcomes

  1. Analyze polynomial expressions to identify potential grouping opportunities.
  2. Identify and extract common factors from grouped terms.
  3. Apply factoring-by-grouping techniques to simplify complex algebraic expressions.
  4. Evaluate the correctness of factored expressions through substitution and reverse factoring methods.
  5. Develop problem-solving strategies for tackling challenging factoring problems in algebra.

Factoring Practice Quiz: Grouping Problems Cheat Sheet

  1. Understand the Basics of Factoring by Grouping - Factoring by grouping splits a polynomial into manageable chunks so you can spot and pull out common elements with ease. By turning ax + ay + bx + by into (ax + ay) + (bx + by), you make the expression far simpler. Embrace this foundational trick to tackle more complex problems with confidence! Factoring by Grouping Tutorial
  2. Identify Common Factors in Each Group - Once you've grouped the terms, scan each pair for its greatest common factor (GCF). For example, (3x³ + 6x²) + (2x + 4) reveals GCFs of 3x² and 2, taking you a big step closer to the final factorization. This clear‑eyed spotting game supercharges your simplification skills. Step-by-Step Factoring Guide
  3. Practice Rearranging Terms for Effective Grouping - If grouping feels awkward, mix up the order of your terms until patterns emerge. Turning 5x³ + 7x² - 20x - 28 into (5x³ + 7x²) - (20x + 28) makes factoring a breeze. Play with term order like a puzzle to find your perfect match! MathBits Grouping Guide
  4. Apply the Distributive Property - After spotting your GCF, use the distributive property to factor it out from each group. For instance, x(x - m) - n(x - m) neatly collapses to (x - m)(x - n). Mastering this move is like having a superpower for simplifying expressions. Distributive Property Walkthrough
  5. Recognize When to Use Factoring by Grouping - Factoring by grouping is your go‑to when a polynomial has four terms or when other methods stall out. Spotting the right moment to apply this strategy saves you time and headaches. Keep an eye out for those four‑term clues and become an efficient problem solver! Factoring by Grouping on Socratic
  6. Check for Further Factorization - After your first round of factoring, pause to see if the results break down even more. Expressions like (x² - 4) can still be factored into (x - 2)(x + 2). This extra check ensures you never miss a simpler form. Advanced Factoring Tips
  7. Practice with Various Polynomials - Boost your confidence by tackling a diverse set of four‑term polynomials until grouping feels like second nature. Regular drills help you recognize common patterns and shortcuts faster. Remember, the more you practice, the less scary it becomes! PO Grouping Practice Problems
  8. Understand the Role of Coefficients - Coefficients can make or break your grouping game, so pay close attention to them. In 2x² + 5x + 3, the numbers guide how you pair and factor terms. Spotting these numerical hints sharpens your overall approach. Coefficient Clues Guide
  9. Learn to Recognize Prime Polynomials - Some polynomials won't factor over the integers - learn to spot these prime cases early. Identifying them quickly prevents wasted effort on impossible groupings. With practice, you'll know when to switch strategies like a factoring ninja! Prime Polynomial Indicators
  10. Utilize Online Resources for Additional Practice - Level up by exploring reputable platforms offering interactive problems and step‑by‑step solutions. Sites like MathBits and Socratic are treasure troves of practice sets and clear explanations. Schedule regular practice sessions to keep your factoring skills sharp! Extra Practice Portal
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