Sum and Difference of Cubes

Show your work for each problem.

1.
Which of the following is a perfect cube?
1. a)
2. b)
3. c)
4. d)
Answer:
. The other numbers are not perfect cubes: , , has no integer cube root.
2.
What is ?
Answer: 4
because .
3.
In the sum of cubes formula , what sign is the middle term of the trinomial?
1. a)Positive (+)
2. b)Zero
3. c)It depends on a and b
4. d)Negative (-)
Answer: Negative (-)
In the sum of cubes, the middle term is (negative). Remember SOAP: Same, Opposite, Always Positive. The middle term has the Opposite sign to the binomial.
4.
Factor
Answer: (x + 2)(x^2 - 2x + 4)
1. What is (the cube root of )? x
2. What is (the cube root of )? 2
3. What is the binomial factor ? (x + 2)
4. What is ? x^2
5. What is ? 2x
6. What is ? 4
5.
Factor . What is the binomial factor?
Answer: (y - 1)
. The binomial factor is .
6.
Factor
Answer: (x - 3)(x^2 + 3x + 9)
1. What is (the cube root of )? x
2. What is (the cube root of )? 3
3. What is the binomial factor ? (x - 3)
4. What is the trinomial factor ? (x^2 + 3x + 9)
7.
What is factored?
1. a)
2. b)
3. c)
4. d)
Answer:
8.
Factor . Write the binomial factor.
Answer: (4 - y)
. Here and .
9.
Factor
Answer: 2(x + 2)(x^2 - 2x + 4)
1. What is the GCF of and ? 2
2. Factor out the GCF. What is inside the parentheses? (x^3 + 8)
3. Now factor using sum of cubes. What is ? 2
4. What is the complete factorization? 2(x + 2)(x^2 - 2x + 4)
10.
Can the trinomial be factored further over the real numbers?
1. a)Yes, as
2. b)Yes, as
3. c)No, it is prime
4. d)Yes, as
Answer: No, it is prime
The discriminant . Since it's negative, the trinomial has no real roots and cannot be factored over the reals.
11.
Factor
Answer: (3a - 2b)(9a^2 + 6ab + 4b^2)
1. What is the cube root of ? 3a
2. What is the cube root of ? 2b
3. What is ? 9a^2
4. What is ? 6ab
5. What is ? 4b^2
12.
Factor . Write the trinomial factor.
Answer: (25x^2 - 20xy + 16y^2)
. The trinomial has , , and .
13.
Factor completely.
1. a)
2. b)
3. c)
4. d)
Answer:
. But is a difference of squares. So the complete factorization is . Note: this can also be factored as using sum/difference of cubes in a different way, giving .
14.
Factor completely.
Answer: 3(x + 3y)(x^2 - 3xy + 9y^2)
1. What is the GCF of and ? 3
2. After factoring out the GCF, what is inside the parentheses? (x^3 + 27y^3)
3. What is the cube root of ? 3y
4. What is the complete factorization? 3(x + 3y)(x^2 - 3xy + 9y^2)
15.
What is ?
Answer: 5
because .
16.
Factor
Answer: (x + 5)(x^2 - 5x + 25)
1. What is the cube root of ? 5
2. What is the binomial factor ? (x + 5)
3. What is ? 25
4. What is the full trinomial factor? (x^2 - 5x + 25)
17.
Factor . Write the binomial factor.
Answer: (10 - 3x)
. The binomial factor is .
18.
Which expression is equivalent to ?
1. a)
2. b)
3. c)
4. d)
Answer:
is the factored form of (sum of cubes).