<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />

# 6.6: Sum and Difference of Cubes

Difficulty Level: At Grade Created by: CK-12
Estimated9 minsto complete
%
Progress
Practice Sum and Difference of Cubes

MEMORY METER
This indicates how strong in your memory this concept is
Progress
Estimated9 minsto complete
%
Estimated9 minsto complete
%
MEMORY METER
This indicates how strong in your memory this concept is

The volume of a rectangular prism is 2x4128x\begin{align*}2x^4 - 128x\end{align*}. What are the lengths of the prism's sides?

### Watch This

First watch this video.

### Guidance

In the previous chapter, you learned how to factor several different types of quadratic equations. Here, we will expand this knowledge to certain types of polynomials. The first is the sum of cubes. The sum of cubes is what it sounds like, the sum of two cube numbers or a3+b3\begin{align*}a^3+b^3\end{align*}. We will use an investigation involving volume to find the factorization of this polynomial.

#### Investigation: Sum of Cubes Formula

1. Pictorially, the sum of cubes looks like this:

Or, we can put one on top of the other.

2. Recall that the formula for volume is length×width×depth\begin{align*}length \times width \times depth\end{align*}. Find the volume of the sum of these two cubes.

V=a3+b3\begin{align*}V = a^3+b^3\end{align*}

3. Now, we will find the volume in a different way. Using the second picture above, we will add in imaginary lines so that these two cubes look like one large prism. Find the volume of this prism.

V=a×a×(a+b)=a2(a+b)\begin{align*}V &= a \times a \times(a+b)\\ &= a^2(a+b)\end{align*}

4. Subtract the imaginary portion on top. In the picture, they are prism 1 and prism 2.

V=a2(a+b)ab(ab)Prism 1+b2(ab)Prism 2\begin{align*}V = a^2(a+b) - \left[\underbrace{ab(a-b)}_{{\color{red}Prism \ 1}} + \underbrace{b^2(a-b)}_{{\color{red}Prism \ 2}}\right]\end{align*}

5. Pull out any common factors within the brackets.

V=a2(a+b)b(ab)[a+b]\begin{align*}V = a^2(a+b)-b(a-b)[a+b]\end{align*}

6. Notice that both terms have a common factor of (a+b)\begin{align*}(a + b)\end{align*}. Pull this out, put it in front, and get rid of the brackets.

V=(a+b)(a2b(ab))\begin{align*}V = (a+b)(a^2-b(a-b))\end{align*}

7. Simplify what is inside the second set of parenthesis.

V=(a+b)(a2ab+b2)\begin{align*}V = (a+b)(a^2-ab+b^2)\end{align*}

In the last step, we found that a3+b3\begin{align*}a^3+b^3\end{align*} factors to (a+b)(a2ab+b2)\begin{align*}(a+b)(a^2-ab+b^2)\end{align*}. This is the Sum of Cubes Formula.

#### Example A

Factor 8x3+27\begin{align*}8x^3+27\end{align*}.

Solution: First, determine if these are “cube” numbers. A cube number has a cube root. For example, the cube root of 8 is 2 because 23=8\begin{align*}2^3 = 8\end{align*}. 33=27,43=64,53=125\begin{align*}3^3 = 27, 4^3 = 64, 5^3 = 125\end{align*}, and so on.

a3a=8x3=(2x)3b3=27=33=2x b=3\begin{align*}a^3 &= 8x^3 = (2x)^3 \qquad b^3 = 27 = 3^3\\ a &= 2x \qquad \qquad \qquad \ b = 3\end{align*}

In the formula, we have:

(a+b)(a2ab+b2)=(2x+3)((2x)2(2x)(3)+32)=(2x+3)(4x26x+9)\begin{align*}(a+b)(a^2-ab+b^2) &= (2x+3)((2x)^2-(2x)(3)+3^2)\\ &= (2x+3)(4x^2-6x+9)\end{align*}

Therefore, 8x3+27=(2x+3)(4x26x+9)\begin{align*}8x^3+27 = (2x+3)(4x^2-6x+9)\end{align*}. The second factored polynomial does not factor any further.

#### Investigation: Difference of Cubes

1. Pictorially, the difference of cubes looks like this:

Imagine the smaller cube is taken out of the larger cube.

2. Recall that the formula for volume is length×width×depth\begin{align*}length \times width \times depth\end{align*}. Find the volume of the difference of these two cubes.

V=a3b3\begin{align*}V = a^3-b^3\end{align*}

3. Now, we will find the volume in a different way. Using the picture here, will add in imaginary lines so that the shape is split into three prisms. Find the volume of prism 1, prism 2, and prism 3.

Prism 1:aa(ab)Prism 2:ab(ab)Prism 3:bb(ab)\begin{align*}& \text{Prism} \ 1: a \cdot a \cdot (a-b)\\ & \text{Prism} \ 2: a \cdot b \cdot (a-b)\\ & \text{Prism} \ 3: b \cdot b \cdot (a-b)\end{align*}

4. Add the volumes together to get the volume of the entire shape.

V=a2(ab)+ab(ab)+b2(ab)\begin{align*}V = a^2(a-b)+ab(a-b)+b^2(a-b)\end{align*}

5. Pull out any common factors and simplify.

V=(ab)(a2+ab+b2)\begin{align*}V = (a-b)(a^2+ab+b^2)\end{align*}

In the last step, we found that a3b3\begin{align*}a^3-b^3\end{align*} factors to (ab)(a2+ab+b2)\begin{align*}(a-b)(a^2+ab+b^2)\end{align*}. This is the Difference of Cubes Formula.

#### Example B

Factor x5125x2\begin{align*}x^5-125x^2\end{align*}.

Solution: First, take out any common factors.

x5125x2=x2(x3125)\begin{align*}x^5-125x^2 = x^2(x^3-125)\end{align*}

What is inside the parenthesis is a difference of cubes. Use the formula.

x5125x2=x2(x3125)=x2(x353)=x2(x5)(x2+5x+25)\begin{align*}x^5-125x^2 &= x^2(x^3-125)\\ &= x^2(x^3-5^3)\\ &= x^2(x-5)(x^2+5x+25)\end{align*}

#### Example C

Find the real-number solutions of x38=0\begin{align*}x^3-8 = 0\end{align*}.

Solution: Factor using the difference of cubes.

x38(x2)(x2+2x+4)x=0=0=2\begin{align*}x^3-8 &= 0\\ (x-2)(x^2+2x+4) &= 0\\ x& = 2\end{align*}

In the last step, we set the first factor equal to zero. The second factor, x2+2x+4\begin{align*}x^2+2x+4\end{align*}, will give imaginary solutions. For both the sum and difference of cubes, this will always happen.

Intro Problem Revisit We need to factor 2x4128x\begin{align*}2x^4 - 128x\end{align*}.

First, take out any common factors.

2x4128x=2x(x364)\begin{align*}2x^4 - 128x = 2x(x^3 - 64)\end{align*}

What is inside the parenthesis is a difference of cubes. Use the Difference of Cubes Formula.

2x(x364)=2x(x343)=2x(x4)(x2+4x+16)\begin{align*}2x(x^3 - 64)\\ &= 2x(x^3 - 4^3)\\ &= 2x(x - 4)(x^2 + 4x + 16)\end{align*}

Therefore, the side lengths of the rectangular prism are 2x\begin{align*}2x\end{align*}, x+4\begin{align*}x +4\end{align*}, and x2+4x+16\begin{align*}x^2 + 4x + 16\end{align*}.

### Guided Practice

Factor using the sum or difference of cubes.

1. x31\begin{align*}x^3-1\end{align*}

2. 3x3+192\begin{align*}3x^3+192\end{align*}

3. 125216x3\begin{align*}125-216x^3\end{align*}

4. Find the real-number solution to 27x3+8=0\begin{align*}27x^3+8=0\end{align*}.

1. Factor using the difference of cubes.

x31=x313=(x1)(x2+x+1)\begin{align*}x^3-1 &= x^3-1^3\\ &= (x-1)(x^2+x+1)\end{align*}

2. Pull out the 3, then factor using the sum of cubes.

3x3+192=3(x3+64)=3(x3+43)=3(x+4)(x24x+16)\begin{align*}3x^3+192 &= 3(x^3+64)\\ &= 3(x^3+4^3)\\ &= 3(x+4)(x^2-4x+16)\end{align*}

3. Factor using the difference of cubes.

125216x3=53(6x)3=(56x)(52+(5)(6x)+(6x)2)=(56x)(25+30x+36x2)\begin{align*}125-216x^3 &= 5^3-(6x)^3\\ &= (5-6x)(5^2+(5)(6x)+(6x)^2)\\ &= (5-6x)(25+30x+36x^2)\end{align*}

4. Factor using the sum of cubes and then solve.

27x3+8(3x)3+23(3x+2)(9x26x+4)x=0=0=0=23\begin{align*}27x^3+8 &= 0\\ (3x)^3+2^3 &=0\\ (3x+2)(9x^2-6x+4) &= 0\\ x &= -\frac{2}{3}\end{align*}

### Vocabulary

Sum of Cubes Formula
a3+b3=(a+b)(a2ab+b2)\begin{align*}a^3+b^3 = (a+b)(a^2-ab+b^2)\end{align*}
Difference of Cubes Formula
a3b3=(ab)(a2+ab+b2)\begin{align*}a^3-b^3 = (a-b)(a^2+ab+b^2)\end{align*}

### Practice

Factor each polynomial by using the sum or difference of cubes.

1. x327\begin{align*}x^3-27\end{align*}
2. 64+x3\begin{align*}64+x^3\end{align*}
3. 32x34\begin{align*}32x^3-4\end{align*}
4. \begin{align*}64x^3+343\end{align*}
5. \begin{align*}512-729x^3\end{align*}
6. \begin{align*}125x^4+8x\end{align*}
7. \begin{align*}648x^3+81\end{align*}
8. \begin{align*}5x^6-135x^3\end{align*}
9. \begin{align*}686x^7-1024x^4\end{align*}

Find the real-number solutions for each equation.

1. \begin{align*}125x^3+1=0\end{align*}
2. \begin{align*}64-729x^3 = 0\end{align*}
3. \begin{align*}8x^4-343x = 0\end{align*}
4. Challenge Find ALL solutions (real and imaginary) for \begin{align*}5x^5+625x^2 = 0\end{align*}.
5. Challenge Find ALL solutions (real and imaginary) for \begin{align*}686x^3+2000 = 0\end{align*}.
6. Real Life Application You have a piece of cardboard that you would like to fold up and make an open (no top) box out of. The dimensions of the cardboard are \begin{align*}36^{\prime\prime} \times 42^{\prime\prime}\end{align*}. Write a factored equation for the volume of this box. Find the volume of the box when \begin{align*}x = 1, 3,\end{align*} and 5.

### Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes

### Vocabulary Language: English

Cubed

The cube of a number is the number multiplied by itself three times. For example, "two-cubed" = $2^3 = 2 \times 2 \times 2 = 8$.

Difference of Cubes Formula

The difference of cubes formula is $a^3-b^3 = (a-b)(a^2+ab+b^2)$.

Sum of Cubes Formula

The sum of cubes formula is $a^3+b^3 = (a+b)(a^2-ab+b^2)$.

Show Hide Details
Description
Difficulty Level:
Authors:
Tags:
Subjects: