Question

1. a) factor each polynomial.

i) $x^{3}-1$
ii) $x^{3}-27$

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 $x^{3}-1$

Here, $a = x$ and $b = 1$. Substitute into the formula:
$x^{3}-1=(x - 1)(x^{2}+x\times1+1^{2})=(x - 1)(x^{2}+x + 1)$

Step3: Factor $x^{3}-27$

Since $27 = 3^{3}$, here $a=x$ and $b = 3$. Substitute into the formula:
$x^{3}-27=(x - 3)(x^{2}+x\times3+3^{2})=(x - 3)(x^{2}+3x + 9)$

Answer:

i) $(x - 1)(x^{2}+x + 1)$
ii) $(x - 3)(x^{2}+3x + 9)$