# How do you find the product (q+r)^2(q-r)?

Jan 20, 2017

${q}^{3} + {q}^{2} r - q {r}^{2} - {r}^{3}$

#### Explanation:

We are asked to find the product of two numbers ${\left(q + r\right)}^{2}$ and $\left(q - r\right)$. The algebraic expressions $\left(q + r\right)$ and $\left(q - r\right)$ reflect the internal structure of each number.
We can think of ${\left(q + r\right)}^{2}$ as equal to $\left(q + r\right)$$\left(q + r\right)$

So ${\left(q + r\right)}^{2}$$\left(q - r\right)$ = $\left(q + r\right)$$\left(q + r\right)$ $\left(q - r\right)$

We should be familiar with the result

$\left(q + r\right)$$\left(q - r\right)$= $\left({q}^{2} - {r}^{2}\right)$

So $\left(q + r\right)$$\left(q + r\right)$ $\left(q - r\right)$= $\left({q}^{2} - {r}^{2}\right)$$\left(q + r\right)$

And multiplying out in full

$\left({q}^{2} - {r}^{2}\right)$$\left(q + r\right)$= ${q}^{3} + {q}^{2} r - q {r}^{2} - {r}^{3}$