# How do you multiply (x-3)^3?

Apr 24, 2017

See the entire solution process below:

#### Explanation:

We can rewrite this expression as:

$\left(x - 3\right) \left(x - 3\right) \left(x - 3\right)$

We can multiple the two terms in parenthesis on the right of the expression using this rule:

$\left(a - b\right) \left(a - b\right) = {a}^{2} - 2 a b + {b}^{2}$

Substituting $x$ for $a$ and $3$ for $b$ gives:

$\left(x - 3\right) \left(x - 3\right) \left(x - 3\right) = \left(x - 3\right) \left({x}^{2} - \left(2 x \cdot 3\right) + 9\right) =$

$\left(x - 3\right) \left({x}^{2} - 6 x + 9\right)$

We now need to multiply these two terms together. To multiply these two terms you multiply each individual term in the left parenthesis by each individual term in the right parenthesis.

$\left(\textcolor{red}{x} - \textcolor{red}{3}\right) \left(\textcolor{b l u e}{{x}^{2}} - \textcolor{b l u e}{6 x} + \textcolor{b l u e}{9}\right)$ becomes:

$\left(\textcolor{red}{x} \times \textcolor{b l u e}{{x}^{2}}\right) - \left(\textcolor{red}{x} \times \textcolor{b l u e}{6 x}\right) + \left(\textcolor{red}{x} \times \textcolor{b l u e}{9}\right) - \left(\textcolor{red}{3} \times \textcolor{b l u e}{{x}^{2}}\right) + \left(\textcolor{red}{3} \times \textcolor{b l u e}{6 x}\right) - \left(\textcolor{red}{3} \times \textcolor{b l u e}{9}\right)$

${x}^{3} - 6 {x}^{2} + 9 x - 3 {x}^{2} + 18 x - 27$

We can now group and combine like terms:

${x}^{3} - 6 {x}^{2} - 3 {x}^{2} + 9 x + 18 x - 27$

${x}^{3} + \left(- 6 - 3\right) {x}^{2} + \left(9 + 18\right) x - 27$

${x}^{3} + \left(- 9\right) {x}^{2} + 27 x - 27$

${x}^{3} - 9 {x}^{2} + 27 x - 27$

Apr 24, 2017

${x}^{3} - 9 {x}^{2} + 27 x - 27$

#### Explanation:

${\left(x - 3\right)}^{3}$ = $\left(x - 3\right) \left(x - 3\right) \left(x - 3\right)$

Taking into account only the first two terms, we multiply to get ${x}^{2} - 6 x + 9$.

Next, we multiply ${x}^{2} - 6 x + 9$ by $x - 3$.

The result is ${x}^{3} - 3 {x}^{2} - 6 {x}^{2} + 18 x + 9 x - 27$.

We simplify this by combining like terms:
${x}^{3} - 9 {x}^{2} + 27 x - 27$