Introduction to Factoring

Show your work for each problem.

1.
Which of the following is a factor of 12?
1. a)4
2. b)5
3. c)9
4. d)7
Answer: 4
$12 \div 4 = 3$ with no remainder, so 4 is a factor of 12. The other numbers (5, 7, 9) do not divide 12 evenly.
2.
What is the GCF of 8 and 12?
Answer: 4
Factors of 8: 1, 2, 4, 8. Factors of 12: 1, 2, 3, 4, 6, 12. The largest common factor is 4.
3.
Factor out the GCF: $4x + 8$
1. a) $2(2x + 4)$
2. b) $4(x + 2)$
3. c) $4(x + 8)$
4. d) $8(x + 1)$
Answer: $4(x + 2)$
The GCF of 4 and 8 is 4. Factoring out 4: $4x + 8 = 4(x + 2)$ . Verify: $4(x + 2) = 4x + 8$ ✓
4.
Factor: $3x + 9$
Answer: 3(x + 3)
1. What is the GCF of 3 and 9? 3
2. After factoring out 3 from $3x$ , what remains? x
3. After factoring out 3 from 9, what remains? 3
4. Which is the factored form of $3x + 9$ ? 3(x + 3)
5.
Factor: $x^2 + 4x$ . Write as $x(\text{something})$ . What goes in the parentheses?
Answer: x + 4
Both $x^2$ and $4x$ contain $x$ . Dividing: $x^2 \div x = x$ and $4x \div x = 4$ . So $x^2 + 4x = x(x + 4)$ .
6.
Factor: $x^2 + 5x + 6$
1. a) $(x + 2)(x + 3)$
2. b) $(x + 5)(x + 1)$
3. c) $(x - 2)(x - 3)$
4. d) $(x + 1)(x + 6)$
Answer: $(x + 2)(x + 3)$
Need numbers with product 6 and sum 5. Try: $2 \times 3 = 6$ and $2 + 3 = 5$ ✓. So $x^2 + 5x + 6 = (x + 2)(x + 3)$ .
7.
Factor: $x^2 + 8x + 15$
Answer: (x + 3)(x + 5)
1. What is the product we need (the constant term)? 15
2. What is the sum we need (the coefficient of $x$ )? 8
3. Which two numbers multiply to 15 and add to 8? 3 and 5
4. Which is the factored form? (x + 3)(x + 5)
8.
Factor $x^2 + 9x + 20$ in the form $(x + a)(x + b)$ with $a < b$ . What is $b$ ?
Answer: 5
Factor pairs of 20: (1, 20), (2, 10), (4, 5). Only $4 + 5 = 9$ . So $x^2 + 9x + 20 = (x + 4)(x + 5)$ , which gives $a = 4$ and $b = 5$ .
9.
Factor out the GCF: $6x^2 + 9x$
1. a) $3x(2x + 3)$
2. b) $6x(x + 9)$
3. c) $3(2x^2 + 3x)$
4. d) $x(6x + 9)$
Answer: $3x(2x + 3)$
GCF of 6 and 9 is 3. GCF of $x^2$ and $x$ is $x$ . So GCF = $3x$ . Factor: $6x^2 + 9x = 3x(2x + 3)$ .
10.
Factor: $x^2 + 6x + 8$
Answer: (x + 2)(x + 4)
1. How many factor pairs of positive whole numbers does 8 have? 2
2. Which pair sums to 6? 2 and 4
3. Which is the factored form? (x + 2)(x + 4)
11.
Factor: $x^2 - 5x + 6$
1. a) $(x - 1)(x - 6)$
2. b) $(x - 2)(x - 3)$
3. c) $(x + 2)(x + 3)$
4. d) $(x + 2)(x - 3)$
Answer: $(x - 2)(x - 3)$
Product = 6 (positive), sum = -5 (negative). Both factors negative: $(-2) \times (-3) = 6$ and $(-2) + (-3) = -5$ . So $(x-2)(x-3)$ .
12.
Factor: $x^2 + x - 12$
Answer: (x + 4)(x - 3)
1. The product is -12 (negative). What does this tell us about the signs? one positive, one negative
2. The sum is +1. Which number should be larger in absolute value? the positive one
3. Which pair of factors of 12 differs by 1? 3 and 4
4. So we need +4 and -3. Which is the factored form? (x + 4)(x - 3)
13.
Factor completely: $2x^2 + 10x + 12$ . The result has the form $a(x + b)(x + c)$ with $b < c$ . What is $c$ ?
Answer: 3
GCF = 2. So $2x^2 + 10x + 12 = 2(x^2 + 5x + 6)$ . Now factor trinomial: $2 \times 3 = 6$ , $2 + 3 = 5$ . Final: $2(x+2)(x+3)$ , so $a = 2$ , $b = 2$ , $c = 3$ .
14.
Factor: $x^2 - 4x - 21$
Answer: (x + 3)(x - 7)
1. What is the product (c)? -21
2. What is the sum (b)? -4
3. Since product is negative, what can we say about the signs? different signs
4. Factor pairs of 21: (1,21), (3,7). Which gives difference of 4? 3 and 7
5. Sum is -4, so the number with the larger absolute value is negative. Which is the factored form? (x + 3)(x - 7)
15.
Which expression is completely factored?
1. a) $3(x+2)(x+4)$
2. b) $3x(x + 6) + 24$
3. c) $3(x^2 + 6x + 8)$
4. d) $(3x+6)(x+4)$
Answer: $3(x+2)(x+4)$
$3(x^2 + 6x + 8)$ is not complete, because $x^2 + 6x + 8 = (x+2)(x+4)$ . $3x(x + 6) + 24$ is a sum, not a product. In $(3x+6)(x+4)$ the factor $3x + 6 = 3(x+2)$ can still be factored. Only $3(x+2)(x+4)$ is completely factored.
16.
Factor out the GCF: $5x^2 - 15x = \square\,(x - 3)$ . What goes in the box?
Answer: 5x
GCF of 5 and 15 is 5. GCF of $x^2$ and $x$ is $x$ . Total GCF = $5x$ . Factor: $5x^2 - 15x = 5x(x - 3)$ .
17.
Find all factor pairs of 18.
Answer: (1,18), (2,9), (3,6)
1. Start with 1. What is 18 divided by 1? 18
2. Try 2. What is 18 divided by 2? 9
3. Try 3. What is 18 divided by 3? 6
4. $\sqrt{18} \approx 4.2$ , so check 4 too: $18 \div 4 = 4.5$ , not a whole number. How many factor pairs does 18 have? 3