Factoring Completely

Show your work for each problem.

1.
What is the first step when factoring any polynomial completely?
1. a)Try grouping
2. b)Factor out the GCF
3. c)Look for difference of squares
4. d)Guess and check
Answer: Factor out the GCF
Always look for a GCF first! This simplifies the polynomial and makes other factoring techniques easier to apply.
2.
Factor completely: . Enter your answer in the form .
Answer: 2(x+2)(x-2)
. First we factor out the GCF of 2, then recognize as a difference of squares.
3.
Which expression is completely factored?
1. a)
2. b)
3. c)
4. d)
Answer:
is completely factored. can still be factored (difference of squares). has a GCF of 3 in the first factor. is not factored at all.
4.
Factor completely:
Answer: 5(x+1)(x+2)
1. What is the GCF of all three terms? 5
2. After factoring out 5, what trinomial remains? x^2 + 3x + 2
3. What two numbers multiply to 2 and add to 3? 1 and 2
4. Write the complete factorization. 5(x+1)(x+2)
5.
Factor completely:
Answer: 3x(x+1)(x-1)
. First factor out the GCF , then recognize as a difference of squares.
6.
Factor completely:
1. a)
2. b)
3. c)
4. d)
Answer:
. Note that is a sum of squares and cannot be factored over real numbers.
7.
Factor completely:
Answer: 2x(x+3)(x-1)
1. What is the GCF of all three terms? 2x
2. After factoring out , what trinomial remains? x^2 + 2x - 3
3. What two numbers multiply to and add to ? 3 and -1
4. Write the complete factorization. 2x(x+3)(x-1)
8.
Factor completely:
Answer: 4(x+3)(x-3)
. Always factor out the GCF first!
9.
Why can't be factored over the real numbers?
1. a)It's already completely factored
2. b)It needs to be grouped first
3. c)It's a sum of squares, which has no real factors
4. d)The GCF is 1
Answer: It's a sum of squares, which has no real factors
A sum of squares () cannot be factored into real number factors. Only the difference of squares () can be factored as .
10.
Factor completely:
Answer: 3(x+2)(x^2-2x+4)
1. What is the GCF? 3
2. After factoring out 3, what expression remains? x^3 + 8
3. What pattern is ? sum of cubes
4. Write the complete factorization using sum of cubes formula. 3(x+2)(x^2-2x+4)
11.
Factor completely:
Answer: 6y(x+2)(x-2)
. The GCF is , and is a difference of squares.
12.
Factor completely:
Answer: 2(x^2+9)(x+3)(x-3)
1. What is the GCF? 2
2. After factoring out 2, what remains? x^4 - 81
3. Recognize the pattern: . Factor this difference of squares. (x^2+9)(x^2-9)
4. Can be factored further? yes
5. Write the complete factorization. 2(x^2+9)(x+3)(x-3)
13.
Factor completely:
Answer: 5(x-2)(x^2+2x+4)
. Using with and .
14.
Factor completely:
1. a)
2. b)
3. c)
4. d)
Answer:
Group: .
15.
Factor completely:
Answer: -3x(x-5)(x+1)
1. What is the GCF (including the negative sign)? -3x
2. After factoring out , what trinomial remains? x^2 - 4x - 5
3. What two numbers multiply to and add to ? -5 and 1
4. Write the complete factorization. -3x(x-5)(x+1)
16.
Factor completely:
Answer: (x+1)(x-1)(x^2+x+1)(x^2-x+1)
. Use difference of squares, then sum and difference of cubes.