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# 7.11: Factorization of Special Cubics

Difficulty Level: Advanced Created by: CK-12
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Practice Sum and Difference of Cubes

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Factor the following cubic polynomial: 375x3+648\begin{align*}375x^3+648\end{align*}.

### Guidance

While many cubics cannot easily be factored, there are two special cases that can be factored quickly. These special cases are the sum of perfect cubes and the difference of perfect cubes.

• Factoring the sum of two cubes follows this pattern: x3+y3=(x+y)(x2xy+y2)\begin{align*}x^3+y^3=(x+y)(x^2-xy+y^2)\end{align*}
• Factoring the difference of two cubes follows this pattern: x3y3=(xy)(x2+xy+y2)\begin{align*}x^3-y^3=(x-y)(x^2+xy+y^2)\end{align*}

#### Example A

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

Solution: This is the sum of two cubes and uses the factoring pattern: x3+y3=(x+y)(x2xy+y2)\begin{align*}x^3+y^3=(x+y)(x^2-xy+y^2)\end{align*}.

x3+33=(x+3)(x23x+9)\begin{align*}x^3+3^3=(x+3)(x^2-3x+9)\end{align*}.

#### Example B

Factor: x3343\begin{align*}x^3-343\end{align*}.

Solution: This is the difference of two cubes and uses the factoring pattern: x3y3=(xy)(x2+xy+y2)\begin{align*}x^3-y^3=(x-y)(x^2+xy+y^2)\end{align*}.

x373=(x7)(x2+7x+49)\begin{align*}x^3-7^3=(x-7)(x^2+7x+49)\end{align*}.

#### Example C

Factor: 64x31\begin{align*}64x^3-1\end{align*}.

Solution: This is the difference of two cubes and uses the factoring pattern: x3y3=(xy)(x2+xy+y2)\begin{align*}x^3-y^3=(x-y)(x^2+xy+y^2)\end{align*}.

(4x)313=(4x1)(16x2+4x+1)\begin{align*}(4x)^3-1^3=(4x-1)(16x^2+4x+1)\end{align*}.

#### Concept Problem Revisited

Factor the following cubic polynomial: 375x3+648\begin{align*}375x^3+648\end{align*}.

First you need to recognize that there is a common factor of 3\begin{align*}3\end{align*}. 375x3+648=3(125x3+216)\begin{align*}375x^3+648=3(125x^3+216)\end{align*}

Notice that the result is the sum of two cubes. Therefore, the factoring pattern is x3+y3=(x+y)(x2xy+y2)\begin{align*}x^3+y^3=(x+y)(x^2-xy+y^2)\end{align*}.

375x3+648=3(5x+6)(25x230x+36)\begin{align*}375x^3 +648 = 3(5x+6)(25x^2-30x+36)\end{align*}

### Vocabulary

Difference of Two Cubes
The difference of two cubes is a special polynomial in the form of x3y3\begin{align*}x^3-y^3\end{align*}. This type of polynomial can be quickly factored using the pattern: (x3y3)=(xy)(x2+xy+y2)\begin{align*}(x^3-y^3)=(x-y)(x^2+xy+y^2)\end{align*}
Sum of Two Cubes
The sum of two cubes is a special polynomial in the form of x3+y3\begin{align*}x^3+y^3\end{align*}. This type of polynomial can be quickly factored using the pattern: (x3+y3)=(x+y)(x2xy+y2)\begin{align*}(x^3+y^3)=(x+y)(x^2-xy+y^2)\end{align*}

### Guided Practice

Factor each of the following cubics.

1. x3+512\begin{align*}x^3+512\end{align*}

2. 8x3+125\begin{align*}8x^3+125\end{align*}

3. x3216\begin{align*}x^3-216\end{align*}

1. x3+83=(x+8)(x28x+64)\begin{align*}x^3+8^3=(x+8)(x^2-8x+64)\end{align*}.

2. (2x)3+53=(2x+5)(4x210x+25)\begin{align*}(2x)^3+5^3=(2x+5)(4x^2-10x+25)\end{align*}.

3. x363=(x6)(x2+6x+36)\begin{align*}x^3-6^3=(x-6)(x^2+6x+36)\end{align*}.

### Practice

Factor each of the following cubics.

1. x3+h3\begin{align*}x^3+h^3\end{align*}
2. a3+125\begin{align*}a^3+125\end{align*}
3. 8x3+64\begin{align*}8x^3+64\end{align*}
4. x3+1728\begin{align*}x^3+1728\end{align*}
5. 2x3+6750\begin{align*}2x^3+6750\end{align*}
6. h364\begin{align*}h^3-64\end{align*}
7. s3216\begin{align*}s^3-216\end{align*}
8. p3512\begin{align*}p^3-512\end{align*}
9. 4e332\begin{align*}4e^3-32\end{align*}
10. 2w3250\begin{align*}2w^3-250\end{align*}
11. x3+8\begin{align*}x^3+8\end{align*}
12. y31\begin{align*}y^3-1\end{align*}
13. 125e38\begin{align*}125e^3-8\end{align*}
14. 64a3+2197\begin{align*}64a^3+2197\end{align*}
15. 54z3+3456\begin{align*}54z^3+3456\end{align*}

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### Vocabulary Language: English

Cubed

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

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