Factoring Quiz
Questions: 16 · 10 minutes
1. What is the greatest-common-factor form of 6x + 9?
9(6x + 1)
6(x + 3)
3(2x + 6)
3(2x + 3)
2. A student must factor 12x^3 - 18x^2 completely. Which answer removes the greatest common factor and leaves no further integer factoring to do?
3x^2(4x - 6)
6x^2(2x - 3)
6x(2x^2 - 3x)
2x^2(6x - 9)
3. Which factorization correctly applies the difference-of-cubes identity to x^3 - 8?
(x - 2)^3, which is a binomial cube rather than a difference-of-cubes factorization
(x - 2)(x^2 - 2x + 4), which uses the wrong sign on the middle term
(x + 2)(x^2 - 2x + 4), which begins with the sum rather than the difference
(x - 2)(x^2 + 2x + 4), following a^3 - b^3 = (a - b)(a^2 + ab + b^2)
4. Which pair of binomials multiplies to 6x^2 + x - 2?
(3x + 2)(2x - 1), because the cross terms combine to x
(6x - 2)(x + 1), because this instead creates a middle term of 4x
(2x + 2)(3x - 1), because this product has a common factor and middle term 4x
(3x - 2)(2x + 1), because its cross terms combine to -x
5. Which is the complete factorization of 4x^2 - 12x + 9?
(2x + 3)^2, which gives a positive middle term
(4x - 3)(x - 3), which gives a leading coefficient of 4 but the wrong middle term
(2x - 3)^2, the square of a binomial with terms 2x and 3
(2x - 9)(2x - 1), which gives the wrong middle coefficient and constant
6. After grouping x^3 + 3x^2 + 2x + 6 as (x^3 + 3x^2) + (2x + 6), what is the complete factorization?
(x + 2)(x^2 + 3), which does not preserve the grouped common binomial
(x + 3)(x^2 - 2), which gives the wrong signs for the last two terms
(x + 3)(x^2 + 2), obtained from x^2(x + 3) + 2(x + 3)
(x - 3)(x^2 + 2), which creates negative terms when expanded
7. What is the complete factorization of 3x^2 - 12 over the integers?
3(x - 4)(x + 1)
3(x - 2)(x + 2)
(3x - 6)(x + 2)
3(x - 2)^2
8. How does x^2 - 25 factor over the integers?
(x - 5)^2
(x - 5)(x + 5)
(x - 25)(x + 1)
(x - 1)(x + 25)
9. A rectangle has area x^2 + 8x + 15 square units. Which pair of expressions can represent its side lengths?
x + 3 and x + 5, whose product has middle term 8x
x + 1 and x + 15, whose product has middle term 16x
x - 3 and x - 5, whose product has a negative middle term
x + 2 and x + 7, whose product has constant term 14
10. Which polynomial cannot be written as a product of two nonconstant polynomials with integer coefficients?
x^2 + 2x + 1
x^2 - 1
x^2 + x + 1
x^2 + 3x + 2
11. You factor x^2 - 5x + 6 = 0 before solving. Which solutions result?
x = 2 or x = 3, from (x - 2)(x - 3) = 0
x = -2 or x = -3, from incorrectly reversing both solution signs
x = 1 or x = 6, using factors of 6 whose sum is 7
x = -1 or x = -6, using the wrong factor pair and signs
12. Which factorization shows that x^2 - 6x + 9 is a perfect-square trinomial?
(x - 9)(x + 1)
(x - 3)^2
(x - 3)(x + 3)
(x - 1)(x - 9)
13. Which factorization expands to 2x^2 + 7x + 3?
(2x + 3)(x + 1), producing a middle term of 5x
(2x - 1)(x - 3), producing negative middle terms
(2x + 1)(x + 3), producing 6x + x in the middle
(2x + 1)(x - 3), producing a negative constant term
14. Which product expands to x^2 + 7x + 12?
(x + 2)(x + 6), which has a middle coefficient of 8
(x + 1)(x + 12), which has a middle coefficient of 13
(x - 3)(x - 4), which produces a positive constant but a negative middle term
(x + 3)(x + 4), whose inner and outer terms sum to 7x
15. A student writes x^2 - 9 = (x - 3)^2. Which explanation best identifies the error?
A difference of squares needs conjugate factors, so the correct result is (x - 3)(x + 3).
The constant 9 must always be separated into the factors 1 and 9.
A common factor of x should be removed from both terms before any other step.
A missing middle term means the expression cannot be factored over the integers.
16. An expression is grouped as (ax + ay) + (bx + by). Which complete factorization follows?
(a + x)(b + y)
ab(x + y)
(a + y)(b + x)
(a + b)(x + y)