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
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:
1. a)
2. b)
3. c)
4. d)
Answer:
The GCF of 4 and 8 is 4. Factoring out 4: . Verify: ✓
4.
Factor:
Answer: 3(x + 3)
1. What is the GCF of 3 and 9? 3
2. After factoring out 3 from , what remains? x
3. After factoring out 3 from 9, what remains? 3
4. Write the factored form: 3(x + 3)
5.
Factor: . Write as . What goes in the parentheses?
Answer: x + 4
Both and contain . Dividing: and . So .
6.
Factor:
1. a)
2. b)
3. c)
4. d)
Answer:
Need numbers with product 6 and sum 5. Try: and ✓. So .
7.
Factor:
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 )? 8
3. Which two numbers multiply to 15 and add to 8? 3 and 5
4. Write the factored form (x + 3)(x + 5)
8.
Factor . Enter your answer in the form with .
Answer: (x+4)(x+5)
Factor pairs of 20: (1,20), (2,10), (4,5). Only 4+5=9. So .
9.
Factor:
1. a)
2. b)
3. c)
4. d)
Answer:
GCF of 6 and 9 is 3. GCF of and is . So GCF = . Factor: .
10.
Factor:
Answer: (x + 2)(x + 4)
1. List the factor pairs of 8 (1,8), (2,4)
2. Which pair sums to 6? 2 and 4
3. Write the factored form (x + 2)(x + 4)
11.
Factor:
1. a)
2. b)
3. c)
4. d)
Answer:
Product = 6 (positive), sum = -5 (negative). Both factors negative: and . So .
12.
Factor:
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. Write the factored form. (x + 4)(x - 3)
13.
Factor completely:
Answer: 2(x + 2)(x + 3)
GCF = 2. So . Now factor trinomial: , . Final: .
14.
Factor:
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 larger number is negative. Write the factored form. (x + 3)(x - 7)
15.
Which expression is completely factored?
1. a)
2. b)
3. c)
4. d)
Answer:
Option B: trinomial can be factored. Option C: this equals , not a standard factored form of the original. Option D: , so it's not fully factored. Only A is completely factored.
16.
Factor:
Answer: 5x(x - 3)
GCF of 5 and 15 is 5. GCF of and is . Total GCF = . Factor: .
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. The square root of 18 is about 4.2. We've checked up to 3. List all pairs. (1,18), (2,9), (3,6)