# How do you multiply (x+4)^3?

Jul 4, 2017

See a solution process below:

#### Explanation:

We can use Pascal's triangle to solve this problem.

The triangle values for the exponent 3 are:

$1 \textcolor{w h i t e}{\ldots \ldots \ldots} 3 \textcolor{w h i t e}{\ldots \ldots \ldots} 3 \textcolor{w h i t e}{\ldots \ldots \ldots} 1$

Therefore ${\left(x + 4\right)}^{3}$ can be multiplied as:

$\left(1 \times {x}^{3}\right) + \left(3 \times 4 {x}^{2}\right) + \left(3 \times {4}^{2} x\right) + \left(1 \times {4}^{3}\right)$

${x}^{3} + 12 {x}^{2} + 48 x + 64$

Jul 4, 2017

${x}^{3} + 12 {x}^{2} + 48 x + 64$

#### Explanation:

$\text{factors of the form}$

$\left(x + a\right) \left(x + b\right) \left(x + c\right) \text{ can be expanded as}$

${x}^{3} + \left(a + b + c\right) {x}^{2} + \left(a b + a c + b c\right) x + a b c$

$\Rightarrow {\left(x + 4\right)}^{3} = \left(x + 4\right) \left(x + 4\right) \left(x + 4\right)$

$\text{with } a = b = c = 4$

$\Rightarrow {\left(x + 4\right)}^{3}$

$= {x}^{3} + \left(4 + 4 + 4\right) {x}^{2} + \left(16 + 16 + 16\right) x + {4}^{3}$

$= {x}^{3} + 12 {x}^{2} + 48 x + 64$