# How do you factor x^4 - 3x^2 - 4?

Apr 5, 2018

See a solution process below:

#### Explanation:

Because the ${x}^{4}$ coefficient is $1$ we know the coefficient for the ${x}^{2}$ terms in the factor will also be $1$:

$\left({x}^{2}\right) \left({x}^{2}\right)$

Because the constant is a negative and the coefficient for the $x$ term is a negative we know the sign for the constants in the factors will have one positive and one negative:

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

Now we need to determine the factors which multiply to -4 and also add to -3:

$1 \times - 4 = - 4$; $1 - 4 = - 3$ <- this IS the factor

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

The factor $\left({x}^{2} - 4\right)$ is a special form of the quadratic:

${\textcolor{red}{x}}^{2} - {\textcolor{b l u e}{y}}^{2} = \left(\textcolor{red}{x} + \textcolor{b l u e}{y}\right) \left(\textcolor{red}{x} - \textcolor{b l u e}{y}\right)$

We can factor this term as:

$\left({x}^{2} + 1\right) \left(\textcolor{red}{x} + \textcolor{b l u e}{2}\right) \left(\textcolor{red}{x} - \textcolor{b l u e}{2}\right)$