Question

unit 7 review - s. o. a. p.
sum and difference of cubes

1. $x^{3}-216$
2. $x^{3}+512$
3. $27x^{3}+64$
4. $4x^{3}-108$
5. $1000x^{3}-1$
6. $8x^{3}+125$

Explanation:

Step1: Recall difference of cubes formula

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

Step2: Factor $x^3 - 216$

Identify $a=x$, $b=6$ (since $6^3=216$):
$x^3 - 216=(x - 6)(x^2 + 6x + 36)$

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Step1: Recall sum of cubes formula

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

Step2: Factor $x^3 + 512$

Identify $a=x$, $b=8$ (since $8^3=512$):
$x^3 + 512=(x + 8)(x^2 - 8x + 64)$

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Step1: Rewrite terms as perfect cubes

$27x^3=(3x)^3$, $64=4^3$

Step2: Apply sum of cubes formula

Identify $a=3x$, $b=4$:
$27x^3 + 64=(3x + 4)(9x^2 - 12x + 16)$

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Step1: Factor out GCF first

Factor 4 from $4x^3 - 108$:
$4(x^3 - 27)$

Step2: Factor the difference of cubes

Identify $a=x$, $b=3$ (since $3^3=27$):
$4(x - 3)(x^2 + 3x + 9)$

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Step1: Rewrite terms as perfect cubes

$1000x^3=(10x)^3$, $1=1^3$

Step2: Apply difference of cubes formula

Identify $a=10x$, $b=1$:
$1000x^3 - 1=(10x - 1)(100x^2 + 10x + 1)$

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Step1: Rewrite terms as perfect cubes

$8x^3=(2x)^3$, $125=5^3$

Step2: Apply sum of cubes formula

Identify $a=2x$, $b=5$:
$8x^3 + 125=(2x + 5)(4x^2 - 10x + 25)$

Answer:

1. $(x - 6)(x^2 + 6x + 36)$
2. $(x + 8)(x^2 - 8x + 64)$
3. $(3x + 4)(9x^2 - 12x + 16)$
4. $4(x - 3)(x^2 + 3x + 9)$
5. $(10x - 1)(100x^2 + 10x + 1)$
6. $(2x + 5)(4x^2 - 10x + 25)$