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Math Foundations · Algebra Toolkit

Factoring Basics — Practice Problems and Worked Solutions

Expose multiplication patterns hidden inside expressions.

The formulas you need here

Difference of two squares

a^{2} - b^{2} = (a - b) (a + b)

What it says: A subtraction of two perfect squares always splits into these two brackets.
When to reach for it: Anything of the form (something)² − (something)², especially inside a fraction you want to cancel.
Watch out: There is no matching rule for a sum: a² + b² does not factor over the real numbers.

\frac{x ^{2} - 9}{x - 3} = \frac{( x - 3 ) ( x + 3 )}{x - 3} = x + 3

This is the cancellation that rescues most 0/0 limits.

17 worked problems

Every step is generated and checked by a computer algebra system.

1.
Choose the correct factorization.
$x^{2} - 1$
Answer: $(x - 1) (x + 1)$
Step-by-step solution
1. $a^{2} - b^{2} = (a - b) (a + b)$
2. $x^{2} - 1 = (x - 1) (x + 1)$
2.
Choose the correct factorization.
$x^{2} - 4$
Answer: $(x - 2) (x + 2)$
Step-by-step solution
1. $a^{2} - b^{2} = (a - b) (a + b)$
2. $x^{2} - 4 = (x - 2) (x + 2)$
3.
Choose the correct factorization.
$x^{2} - 9$
Answer: $(x - 3) (x + 3)$
Step-by-step solution
1. $a^{2} - b^{2} = (a - b) (a + b)$
2. $x^{2} - 9 = (x - 3) (x + 3)$
4.
Choose the correct factorization.
$x^{2} - 16$
Answer: $(x - 4) (x + 4)$
Step-by-step solution
1. $a^{2} - b^{2} = (a - b) (a + b)$
2. $x^{2} - 16 = (x - 4) (x + 4)$
5.
Choose the correct factorization.
$x^{2} - 25$
Answer: $(x - 5) (x + 5)$
Step-by-step solution
1. $a^{2} - b^{2} = (a - b) (a + b)$
2. $x^{2} - 25 = (x - 5) (x + 5)$
6.
Choose the correct factorization.
$x^{2} - 36$
Answer: $(x - 6) (x + 6)$
Step-by-step solution
1. $a^{2} - b^{2} = (a - b) (a + b)$
2. $x^{2} - 36 = (x - 6) (x + 6)$
7.
Choose the correct factorization.
$x^{2} - 49$
Answer: $(x - 7) (x + 7)$
Step-by-step solution
1. $a^{2} - b^{2} = (a - b) (a + b)$
2. $x^{2} - 49 = (x - 7) (x + 7)$
8.
Choose the correct factorization.
$x^{2} - 64$
Answer: $(x - 8) (x + 8)$
Step-by-step solution
1. $a^{2} - b^{2} = (a - b) (a + b)$
2. $x^{2} - 64 = (x - 8) (x + 8)$
9.
Choose the correct factorization.
$x^{2} - 81$
Answer: $(x - 9) (x + 9)$
Step-by-step solution
1. $a^{2} - b^{2} = (a - b) (a + b)$
2. $x^{2} - 81 = (x - 9) (x + 9)$
10.
Choose the correct factorization.
$x^{2} - 100$
Answer: $(x - 10) (x + 10)$
Step-by-step solution
1. $a^{2} - b^{2} = (a - b) (a + b)$
2. $x^{2} - 100 = (x - 10) (x + 10)$
11.
Choose the correct factorization.
$x^{2} - 121$
Answer: $(x - 11) (x + 11)$
Step-by-step solution
1. $a^{2} - b^{2} = (a - b) (a + b)$
2. $x^{2} - 121 = (x - 11) (x + 11)$
12.
Choose the correct factorization.
$x^{2} - 144$
Answer: $(x - 12) (x + 12)$
Step-by-step solution
1. $a^{2} - b^{2} = (a - b) (a + b)$
2. $x^{2} - 144 = (x - 12) (x + 12)$
13.
Choose the correct factorization.
$x^{2} - 169$
Answer: $(x - 13) (x + 13)$
Step-by-step solution
1. $a^{2} - b^{2} = (a - b) (a + b)$
2. $x^{2} - 169 = (x - 13) (x + 13)$
14.
Choose the correct factorization.
$x^{2} - 196$
Answer: $(x - 14) (x + 14)$
Step-by-step solution
1. $a^{2} - b^{2} = (a - b) (a + b)$
2. $x^{2} - 196 = (x - 14) (x + 14)$
15.
Choose the correct factorization.
$x^{2} - 225$
Answer: $(x - 15) (x + 15)$
Step-by-step solution
1. $a^{2} - b^{2} = (a - b) (a + b)$
2. $x^{2} - 225 = (x - 15) (x + 15)$
16.
Choose the correct factorization.
$x^{2} - 256$
Answer: $(x - 16) (x + 16)$
Step-by-step solution
1. $a^{2} - b^{2} = (a - b) (a + b)$
2. $x^{2} - 256 = (x - 16) (x + 16)$
17.
Choose the correct factorization.
$x^{2} - 400$
Answer: $(x - 20) (x + 20)$
Step-by-step solution
1. $a^{2} - b^{2} = (a - b) (a + b)$
2. $x^{2} - 400 = (x - 20) (x + 20)$

The formulas you need here

Difference of two squares

17 worked problems

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