Question

1. $729x^{3}+y^{3}$
2. $343x^{3}+64$
3. $5x^{3}-320$
4. $3x^{3}-24$

Explanation:

Step1: Factor sum of cubes (7)

Use sum of cubes formula $a^3+b^3=(a+b)(a^2-ab+b^2)$, where $a=9x$ ($(9x)^3=729x^3$), $b=y$
$729x^3+y^3=(9x)^3+y^3=(9x+y)((9x)^2-(9x)(y)+y^2)$

Step2: Simplify the quadratic term (7)

$(9x)^2-(9x)(y)+y^2=81x^2-9xy+y^2$

Step3: Factor sum of cubes (8)

Use sum of cubes formula, where $a=7x$ ($(7x)^3=343x^3$), $b=4$ ($4^3=64$)
$343x^3+64=(7x)^3+4^3=(7x+4)((7x)^2-(7x)(4)+4^2)$

Step4: Simplify the quadratic term (8)

$(7x)^2-(7x)(4)+4^2=49x^2-28x+16$

Step5: Factor out GCF first (9)

Factor 5 from $5x^3-320$
$5x^3-320=5(x^3-64)$

Step6: Factor difference of cubes (9)

Use difference of cubes formula $a^3-b^3=(a-b)(a^2+ab+b^2)$, where $a=x$, $b=4$ ($4^3=64$)
$5(x^3-64)=5(x-4)(x^2+4x+16)$

Step7: Factor out GCF first (10)

Factor 3 from $3x^3-24$
$3x^3-24=3(x^3-8)$

Step8: Factor difference of cubes (10)

Use difference of cubes formula, where $a=x$, $b=2$ ($2^3=8$)
$3(x^3-8)=3(x-2)(x^2+2x+4)$

Answer:

1. $\boldsymbol{(9x+y)(81x^2-9xy+y^2)}$
2. $\boldsymbol{(7x+4)(49x^2-28x+16)}$
3. $\boldsymbol{5(x-4)(x^2+4x+16)}$
4. $\boldsymbol{3(x-2)(x^2+2x+4)}$