# How do you multiply (x - 1)(x + 1)(x - 3)(x + 3)?

May 8, 2016

$\left(x - 1\right) \left(x + 1\right) \left(x - 3\right) \left(x + 3\right) = {x}^{4} - 10 {x}^{2} + 9$

#### Explanation:

Note the difference of squares identity:

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

So we find:

$\left(x - 1\right) \left(x + 1\right) = {x}^{2} - {1}^{2} = {x}^{2} - 1$

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

Then if it helps, we can use FOIL to multiply out the resulting two binomials:

$\left({x}^{2} - 1\right) \left({x}^{2} - 9\right)$

$= {\overbrace{\left({x}^{2} \cdot {x}^{2}\right)}}^{\text{First" + overbrace(((x^2) * (-9)))^"Outside" + overbrace(((-1) * (x^2)))^"Inside" + overbrace(((-1) * (-9)))^"Last}}$

$= {x}^{4} - 9 {x}^{2} - {x}^{2} + 9$

$= {x}^{4} - \left(9 + 1\right) {x}^{2} + 9$

$= {x}^{4} - 10 {x}^{2} + 9$