Algebra 2 Factoring Quiz
Questions: 16 · 10 minutes
1. A learner identifies x³ + 27 as a sum of cubes. Which factorization is correct?
(x + 3)(x² - 3x + 9), following the sum-of-cubes pattern
(x - 3)(x² + 3x + 9), which is the pattern for x³ - 27
(x + 3)(x² + 3x + 9), which uses the wrong sign in the quadratic factor
(x + 3)(x² - 9), which does not expand to the original expression
2. A student needs to factor 6x² + 11x + 3. Which factorization should the student use?
(3x + 3)(2x + 1)
(6x + 1)(x + 3)
(3x + 1)(2x + 3)
(6x + 3)(x + 1)
3. Factor x³ + 3x² + 2x + 6 by grouping.
(x + 2)(x² + 3), whose expansion has different middle terms
(x + 3)(x² - 2), which gives negative terms after expansion
x(x + 3)(x + 2), which introduces an extra x² term
(x + 3)(x² + 2), obtained from x²(x + 3) + 2(x + 3)
4. A student is asked to factor x² + 4x + 7 over the integers. Which conclusion is correct?
It factors as (x + 1)(x + 7), although that product has a middle coefficient of 8
It is prime over the integers because no integer pair has product 7 and sum 4
It factors as (x + 2)(x + 3), although that product has constant term 6
It factors as (x - 1)(x - 7), although that product has a negative middle coefficient
5. Which factorization shows that 4x² - 12x + 9 is a perfect-square trinomial?
(2x - 9)(2x - 1), which has the wrong middle term
(4x - 3)(x - 3), which produces a middle coefficient of -15
(2x - 3)², which matches the pattern a² - 2ab + b²
(2x + 3)², which produces a positive middle term
6. The trinomial x² + kx + 24 factors into two binomials whose positive integer constant terms differ by 2. What is k?
k = 14, from using 2 and 12 even though they differ by 10
k = 8, which does not come from a positive integer factor pair of 24
k = 11, from using 3 and 8 even though their product is 24 but their difference is 5
k = 10, from the constants 4 and 6
7. A quadratic equation is written as 2x² - 7x + 3 = 0. Which pair gives all solutions found by factoring?
x = -1/2 and x = -3, from reversing both required signs
x = 1 and x = 3/2, from an incorrect factor pair
x = -1/2 and x = 3, with one root assigned the wrong sign
x = 1/2 and x = 3, obtained from (2x - 1)(x - 3) = 0
8. How does 9x² - 16 factor over the integers?
(9x - 4)(x + 4)
(3x - 4)²
(3x - 4)(3x + 4)
(9x - 16)(x + 1)
9. Let u = x². After factoring u² - 5u + 4 and then factoring any differences of squares, what is the complete integer factorization of x⁴ - 5x² + 4?
(x² + 1)(x² + 4), which corresponds to a positive middle term
(x² - 1)(x² + 4), which produces 3x² rather than -5x²
(x - 1)(x + 1)(x - 2)(x + 2), the fully factored form
(x - 1)(x - 4)(x² + 1), which does not reproduce the even-powered polynomial
10. Which expression is the complete factorization of x² + 7x + 10?
(x + 1)(x + 10), which produces a middle coefficient of 11
(x + 2)(x + 5), whose constants have sum 7 and product 10
(x - 2)(x - 5), which produces a negative middle term
(x + 3)(x + 4), which produces a constant term of 12
11. Which option gives the complete factorization of 12x³ - 27x over the integers?
3x(4x² - 9), which still contains a difference of squares
3x(2x - 3)(2x + 3), which is completely factored
3x(2x - 3)², which produces a nonzero x² term
x(3x - 3)(4x + 9), which does not expand to the original expression
12. What is the complete factorization of 12x³ - 18x² using the greatest common factor?
2x²(6x - 9), which leaves a common factor of 3 inside
3x²(4x - 6), which leaves a common factor of 2 inside
6x(2x² - 3x), which leaves another factor of x inside
6x²(2x - 3), which removes the full greatest common factor
13. Which expression correctly factors 8y³ - 1 as a difference of cubes?
(2y - 1)(4y² - 2y + 1), with the wrong sign on the middle quadratic term
(2y - 1)(4y² + 2y + 1), following the difference-of-cubes pattern
(2y + 1)(4y² - 2y + 1), beginning with the factor for a sum of cubes
(8y - 1)(y² + y + 1), using an incorrect cube root for 8y³
14. A student writes 4x² - 25 = (4x - 5)(x + 5). Which correction properly uses the difference-of-squares pattern?
(2x - 5)(2x + 5)
(2x - 5)²
(2x - 25)(2x + 1)
(4x - 5)(4x + 5)
15. A rectangle has area x² + 7x + 12 square units and length x + 3 units. Assuming exact polynomial dimensions, which expression gives its width?
x + 4, because the area factors as (x + 3)(x + 4)
x + 3, which would make the area x² + 6x + 9
x + 9, which would make the area x² + 12x + 27
x² + 4, which would produce a cubic area when multiplied by the length
16. For what value of k is x - 2 a factor of x³ + kx² - 5x + 6?
k = -1, because substituting x = 2 makes the polynomial equal zero
k = 1, which makes the polynomial's value at x = 2 equal 8
k = -2, which makes the polynomial's value at x = 2 equal -4
k = 2, which makes the polynomial's value at x = 2 equal 12