Sum and Difference of Cubes

Show your work for each problem.

1.
Which of the following is a perfect cube?
1. a) $50$
2. b) $16$
3. c) $27$
4. d) $36$
Answer: $27$
$27 = 3^3 = 3 \times 3 \times 3$ . The other numbers are not perfect cubes: $16 = 2^4$ , $36 = 6^2$ , $50$ has no integer cube root.
2.
What is $\sqrt[3]{64}$ ?
Answer: 4
$\sqrt[3]{64} = 4$ because $4^3 = 4 \times 4 \times 4 = 64$ .
3.
In the sum of cubes formula $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ , which sign stands in front of the middle term $ab$ of the trinomial?
1. a)Plus (+)
2. b)There is no middle term
3. c)It changes from problem to problem
4. d)Minus (-)
Answer: Minus (-)
In the sum of cubes, the trinomial is $a^2 - ab + b^2$ , so a minus sign stands in front of $ab$ . Remember SOAP: Same, Opposite, Always Positive. The middle term has the Opposite sign to the binomial.
4.
Factor $x^3 + 8$
Answer: (x + 2)(x^2 - 2x + 4)
1. What is $a$ (the cube root of $x^3$ )? x
2. What is $b$ (the cube root of $8$ )? 2
3. What is the binomial factor $(a + b)$ ? (x + 2)
4. What is $a^2$ ? x^2
5. What is $ab$ ? 2x
6. What is $b^2$ ? 4
5.
Factor $y^3 - 1$ . What is the binomial factor?
Answer: (y - 1)
$y^3 - 1 = y^3 - 1^3 = (y - 1)(y^2 + y + 1)$ . The binomial factor is $(y - 1)$ .
6.
Factor $x^3 - 27$
Answer: (x - 3)(x^2 + 3x + 9)
1. What is $a$ (the cube root of $x^3$ )? x
2. What is $b$ (the cube root of $27$ )? 3
3. What is the binomial factor $(a - b)$ ? (x - 3)
4. What is the trinomial factor $(a^2 + ab + b^2)$ ? (x^2 + 3x + 9)
7.
What is $8x^3 + 27$ factored?
1. a) $(2x - 3)(4x^2 + 6x + 9)$
2. b) $(2x + 3)(4x^2 - 6x + 9)$
3. c) $(2x + 3)(2x^2 - 3x + 9)$
4. d) $(2x + 3)(4x^2 + 6x + 9)$
Answer: $(2x + 3)(4x^2 - 6x + 9)$
$8x^3 + 27 = (2x)^3 + 3^3 = (2x + 3)((2x)^2 - (2x)(3) + 3^2) = (2x + 3)(4x^2 - 6x + 9)$
8.
Factor $64 - y^3$ . What is the binomial factor $(a - b)$ ?
Answer: (4 - y)
$64 - y^3 = 4^3 - y^3 = (4 - y)(16 + 4y + y^2)$ . Here $a = 4$ and $b = y$ .
9.
Factor $2x^3 + 16$
Answer: 2(x + 2)(x^2 - 2x + 4)
1. What is the GCF of $2x^3$ and $16$ ? 2
2. Factor out the GCF. What is inside the parentheses? (x^3 + 8)
3. Now factor $x^3 + 8$ using sum of cubes. What is $b$ ? 2
4. What is the complete factorization? 2(x + 2)(x^2 - 2x + 4)
10.
Can the trinomial $x^2 - 3x + 9$ be factored further over the real numbers?
1. a)Yes, as $(x - 1)(x - 9)$
2. b)Yes, as $(x - 3)(x + 3)$
3. c)No, it cannot be factored
4. d)Yes, as $(x - 3)^2$
Answer: No, it cannot be factored
The discriminant $(-3)^2 - 4(1)(9) = 9 - 36 = -27 < 0$ . Since it's negative, the trinomial has no real roots and cannot be factored over the reals.
11.
Factor $27a^3 - 8b^3$
Answer: (3a - 2b)(9a^2 + 6ab + 4b^2)
1. What is the cube root of $27a^3$ ? 3a
2. What is the cube root of $8b^3$ ? 2b
3. What is $(3a)^2$ ? 9a^2
4. What is $(3a)(2b)$ ? 6ab
5. What is $(2b)^2$ ? 4b^2
12.
Factor $125x^3 + 64y^3$ . Write the trinomial factor.
Answer: (25x^2 - 20xy + 16y^2)
$125x^3 + 64y^3 = (5x)^3 + (4y)^3 = (5x + 4y)(25x^2 - 20xy + 16y^2)$ . The trinomial has $(5x)^2 = 25x^2$ , $-(5x)(4y) = -20xy$ , and $(4y)^2 = 16y^2$ .
13.
Factor $x^6 - 64$ completely.
1. a) $(x^2 - 4)(x^4 + 4x^2 + 16)$
2. b) $(x^2 - 4)(x^2 + 2x + 4)(x^2 - 2x + 4)$
3. c) $(x - 2)(x + 2)(x^4 + 4x^2 + 16)$
4. d) $(x - 2)(x + 2)(x^2 + 2x + 4)(x^2 - 2x + 4)$
Answer: $(x - 2)(x + 2)(x^2 + 2x + 4)(x^2 - 2x + 4)$
Start with the difference of squares: $x^6 - 64 = (x^3)^2 - 8^2 = (x^3 - 8)(x^3 + 8)$ . Then use the cube formulas: $x^3 - 8 = (x - 2)(x^2 + 2x + 4)$ and $x^3 + 8 = (x + 2)(x^2 - 2x + 4)$ . Complete factorization: $(x - 2)(x + 2)(x^2 + 2x + 4)(x^2 - 2x + 4)$ . If you start with the difference of cubes instead, you get $(x^2 - 4)(x^4 + 4x^2 + 16)$ , and both factors still factor: $x^2 - 4 = (x - 2)(x + 2)$ and $x^4 + 4x^2 + 16 = (x^2 + 2x + 4)(x^2 - 2x + 4)$ . The other choices are not complete.
14.
Factor $3x^3 + 81y^3$ completely.
Answer: 3(x + 3y)(x^2 - 3xy + 9y^2)
1. What is the GCF of $3x^3$ and $81y^3$ ? 3
2. After factoring out the GCF, what is inside the parentheses? (x^3 + 27y^3)
3. What is the cube root of $27y^3$ ? 3y
4. What is the complete factorization? 3(x + 3y)(x^2 - 3xy + 9y^2)
15.
What is $\sqrt[3]{125}$ ?
Answer: 5
$\sqrt[3]{125} = 5$ because $5^3 = 5 \times 5 \times 5 = 125$ .
16.
Factor $x^3 + 125$
Answer: (x + 5)(x^2 - 5x + 25)
1. What is the cube root of $125$ ? 5
2. What is the binomial factor $(a + b)$ ? (x + 5)
3. What is $b^2$ ? 25
4. What is the full trinomial factor? (x^2 - 5x + 25)
17.
Factor $1000 - 27x^3$ . What is the binomial factor $(a - b)$ ?
Answer: (10 - 3x)
$1000 - 27x^3 = 10^3 - (3x)^3 = (10 - 3x)(100 + 30x + 9x^2)$ . The binomial factor is $(10 - 3x)$ .
18.
Which expression is equivalent to $(x + 2)(x^2 - 2x + 4)$ ?
1. a) $x^3 + 8$
2. b) $x^3 - 8$
3. c) $x^3 + 4$
4. d) $x^3 - 4$
Answer: $x^3 + 8$
$(x + 2)(x^2 - 2x + 4)$ is the factored form of $x^3 + 2^3 = x^3 + 8$ (sum of cubes).