Factoring — Drills
Factoring has 10 drill skills. Each one generates fresh questions every time, with a hint and full worked steps, so you can keep practising a single skill until it settles.
What these drills 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.
Gcf
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Factor 21x + 24.
Worked answer
3(7x + 8)
GCF is 3.
3(7x + 8)
Problem 2
Factor 35x + 56.
Worked answer
7(5x + 8)
GCF is 7.
7(5x + 8)
Trinomial
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Factor x2 - 8x + 12.
Worked answer
(x - 2)(x - 6)
-2 and -6: product 12, sum -8 ✓
(x - 2)(x - 6)
Problem 2
Factor x2 + 2x - 8.
Worked answer
(x + 4)(x - 2)
4 and -2: product -8, sum 2 ✓
(x + 4)(x - 2)
Squares
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Factor 64x2 - 64.
Worked answer
(8x - 8)(8x + 8)
(8x)2 - 82
(8x - 8)(8x + 8)
Problem 2
Factor 1x2 - 36.
Worked answer
(1x - 6)(1x + 6)
(1x)2 - 62
(1x - 6)(1x + 6)
Gcffirst
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Factor 4x2 + 44x + 112 completely.
Worked answer
4(x + 4)(x + 7)
4(x2 + 11x + 28)
4 and 7 work.
4(x + 4)(x + 7)
Problem 2
Factor 5x2 + 35x + 60 completely.
Worked answer
5(x + 3)(x + 4)
5(x2 + 7x + 12)
3 and 4 work.
5(x + 3)(x + 4)
Lead
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Factor 4x2 + 12x + 8.
Worked answer
(4x + 4)(x + 2)
Need two numbers multiplying to 32, adding to 12: 8 and 4
4x2 + 8x + 4x + 8
4x(x + 2) + 4(x + 2)
(4x + 4)(x + 2)
Problem 2
Factor 2x2 + 15x + 18.
Worked answer
(2x + 3)(x + 6)
Need two numbers multiplying to 36, adding to 15: 12 and 3
2x2 + 12x + 3x + 18
2x(x + 6) + 3(x + 6)
(2x + 3)(x + 6)
Perfect
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Factor x2 + 22x + 121.
Worked answer
(x + 11)^2
121 = 112 and 2 × 11 = 22 ✓
(x + 11)2
Problem 2
Factor x2 - 6x + 9.
Worked answer
(x - 3)^2
9 = 32 and 2 × 3 = 6 ✓
(x - 3)2
Prime
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Which expression will not factor over the integers?
- x2 - 49
- x2 + 49
- x2 - 14x + 49
- x2 + 14x + 49
Worked answer
x2 + 49
x2 + 49 is a sum of squares.
It has no real factors.
The others are a difference of squares or a perfect square trinomial.
Grouping
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Factor x3 + 5x2 + 6x + 30 by grouping.
Worked answer
(x + 5)(x^2 + 6)
x2(x + 5) + 6(x + 5)
The bracket (x + 5) is shared.
(x + 5)(x2 + 6)
Problem 2
Factor x3 + 7x2 + 4x + 28 by grouping.
Worked answer
(x + 7)(x^2 + 4)
x2(x + 7) + 4(x + 7)
The bracket (x + 7) is shared.
(x + 7)(x2 + 4)
Completing
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Factor 4x3 - 144x completely.
Worked answer
4x(x - 6)(x + 6)
4x(x2 - 36)
4x(x - 6)(x + 6)
Problem 2
Factor 5x3 - 80x completely.
Worked answer
5x(x - 4)(x + 4)
5x(x2 - 16)
5x(x - 4)(x + 4)
Area
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A rectangle has area x2 + 15x + 56. Give expressions for its two sides.
Worked answer
x + 8 and x + 7
8 and 7: product 56, sum 15
(x + 8)(x + 7)
Problem 2
A rectangle has area x2 + 8x + 12. Give expressions for its two sides.
Worked answer
x + 2 and x + 6
2 and 6: product 12, sum 8
(x + 2)(x + 6)