Question

1. $x^{3}+27$
2. $x^{3}-125$
3. $125x^{3}-y^{3}$
4. $27x^{3}+64$
5. $x^{3}-216y^{3}$
6. $27x^{3}+y^{3}$
7. $3x^{4}+192x$
8. $16x^{5}-54x^{2}$

Explanation:

Step1: Sum of cubes formula: $a^3+b^3=(a+b)(a^2-ab+b^2)$; Difference of cubes formula: $a^3-b^3=(a-b)(a^2+ab+b^2)$

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Problem 1: $x^3+27$

Step1: Rewrite as sum of cubes

$x^3+3^3$

Step2: Apply sum of cubes formula

$(x+3)(x^2-3x+9)$

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Problem 3: $x^3-125$

Step1: Rewrite as difference of cubes

$x^3-5^3$

Step2: Apply difference of cubes formula

$(x-5)(x^2+5x+25)$

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Problem 5: $125x^3-y^3$

Step1: Rewrite as difference of cubes

$(5x)^3-y^3$

Step2: Apply difference of cubes formula

$(5x-y)(25x^2+5xy+y^2)$

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Problem 7: $27x^3+64$

Step1: Rewrite as sum of cubes

$(3x)^3+4^3$

Step2: Apply sum of cubes formula

$(3x+4)(9x^2-12x+16)$

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Problem 9: $x^3-216y^3$

Step1: Rewrite as difference of cubes

$x^3-(6y)^3$

Step2: Apply difference of cubes formula

$(x-6y)(x^2+6xy+36y^2)$

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Problem 11: $27x^3+y^3$

Step1: Rewrite as sum of cubes

$(3x)^3+y^3$

Step2: Apply sum of cubes formula

$(3x+y)(9x^2-3xy+y^2)$

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Problem 13: $3x^4+192x$

Step1: Factor out GCF $3x$

$3x(x^3+64)$

Step2: Rewrite as sum of cubes

$3x(x^3+4^3)$

Step3: Apply sum of cubes formula

$3x(x+4)(x^2-4x+16)$

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Problem 15: $16x^5-54x^2$

Step1: Factor out GCF $2x^2$

$2x^2(8x^3-27)$

Step2: Rewrite as difference of cubes

$2x^2((2x)^3-3^3)$

Step3: Apply difference of cubes formula

$2x^2(2x-3)(4x^2+6x+9)$

Answer:

1. $\boldsymbol{(x+3)(x^2-3x+9)}$
2. $\boldsymbol{(x-5)(x^2+5x+25)}$
3. $\boldsymbol{(5x-y)(25x^2+5xy+y^2)}$
4. $\boldsymbol{(3x+4)(9x^2-12x+16)}$
5. $\boldsymbol{(x-6y)(x^2+6xy+36y^2)}$
6. $\boldsymbol{(3x+y)(9x^2-3xy+y^2)}$
7. $\boldsymbol{3x(x+4)(x^2-4x+16)}$
8. $\boldsymbol{2x^2(2x-3)(4x^2+6x+9)}$