Factoring — Practice
Factoring is multiplication run backwards. Always pull out the common factor first — it makes everything after it easier.
What these questions cover
These questions use 7 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Integer — A whole number, positive, negative, or zero.
- Area — Space inside, measured in square units.
- Perfect square — A number that is some whole number squared: 1, 4, 9, 16, 25, …
- Factor — A number that divides evenly into another.
- GCF — Greatest common factor — the largest number dividing both.
- Difference of squares — a2 - b2 = (a - b)(a + b).
- Perfect square trinomial — a2 ± 2ab + b2 = (a ± b)2.
Problem 1
Factor 6x + 15.
Worked answer
3(2x + 5)
GCF is 3
3(2x + 5)
Problem 2
Factor 4x2 - 10x.
Worked answer
2x(2x - 5)
GCF is 2x
2x(2x - 5)
Problem 3
Factor x2 + 7x + 10.
Worked answer
(x + 2)(x + 5)
2 and 5
(x + 2)(x + 5)
Problem 4
Factor x2 - 9x + 20.
Worked answer
(x - 4)(x - 5)
-4 and -5: product 20, sum -9
(x - 4)(x - 5)
Problem 5
Factor x2 - 36.
Worked answer
(x - 6)(x + 6)
x2 - 62
(x - 6)(x + 6)
Problem 6
Factor x2 + 2x - 15.
Worked answer
(x + 5)(x - 3)
5 and -3: product -15, sum 2
(x + 5)(x - 3)
Problem 7
Factor 25x2 - 49.
Worked answer
(5x - 7)(5x + 7)
(5x)2 - 72
(5x - 7)(5x + 7)
Problem 8
Factor 2x2 + 10x + 12 completely.
Worked answer
2(x + 2)(x + 3)
2(x2 + 5x + 6)
2(x + 2)(x + 3)
Problem 9
Factor 2x2 + 7x + 3.
Worked answer
(2x + 1)(x + 3)
6 and 1
2x2 + 6x + x + 3
2x(x+3) + 1(x+3) = (2x+1)(x+3)
Problem 10
Factor x2 - 14x + 49.
Worked answer
(x - 7)^2
49 = 72 and 2 × 7 = 14 ✓
(x - 7)2
Problem 11
Which expression will not factor over the integers?
- x2 + 9
- x2 - 9
- x2 + 6x + 9
- x2 - 6x + 9
Worked answer
x2 + 9
x2 + 9 is a sum of squares
It has no real factors
Problem 12
Factor x3 + 3x2 + 2x + 6 by grouping.
Worked answer
(x + 3)(x^2 + 2)
x2(x + 3) + 2(x + 3)
(x + 3)(x2 + 2)
Problem 13
Factor 6x2 - 7x - 3.
Worked answer
(3x + 1)(2x - 3)
-9 and 2
6x2 - 9x + 2x - 3
3x(2x - 3) + 1(2x - 3) = (3x+1)(2x-3)
Problem 14
Factor 3x3 - 27x completely.
Worked answer
3x(x - 3)(x + 3)
3x(x2 - 9)
3x(x - 3)(x + 3)
Problem 15
Factor x4 - 81 completely.
Worked answer
(x - 3)(x + 3)(x^2 + 9)
(x2 - 9)(x2 + 9)
(x-3)(x+3)(x2+9)
Problem 16
A rectangle has area x2 + 11x + 24. Give expressions for its two sides.
Worked answer
x + 3 and x + 8
3 and 8: product 24, sum 11
(x + 3)(x + 8)