# Factoring Calculator

The calculator will try to factor any expression (polynomial, binomial, trinomial, quadratic, rational, irrational, exponential, trigonometric, or a mix of them), with steps shown. To do this, some substitutions are first applied to convert the expression into a polynomial, and then the following techniques are used: factoring monomials (common factor), factoring quadratics, grouping and regrouping, square of sum/difference, cube of sum/difference, difference of squares, sum/difference of cubes, and the rational zeros theorem.

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## Solution

Your input: factor $x^{4} - 20 x^{2} + 64$.

We can treat $x^{4} - 20 x^{2} + 64$ as a quadratic function with respect to $x^{2}$.

Let $Y = x^{2}$.

Temporarily rewrite $x^{4} - 20 x^{2} + 64$ in terms of $Y$: $x^{4} - 20 x^{2} + 64$ becomes $Y^{2} - 20 Y + 64$.

To factor the quadratic function $Y^{2} - 20 Y + 64$, we should solve the corresponding quadratic equation $Y^{2} - 20 Y + 64=0$.

Indeed, if $Y_1$ and $Y_2$ are the roots of the quadratic equation $aY^2+bY+c=0$, then $aY^2+bY+c=a(Y-Y_1)(Y-Y_2)$.

Solve the quadratic equation $Y^{2} - 20 Y + 64=0$.

The roots are $Y_{1} = 16$, $Y_{2} = 4$ (use the quadratic equation calculator to see the steps).

Therefore, $Y^{2} - 20 Y + 64 = \left(Y - 16\right) \left(Y - 4\right)$.

Recall that $Y = x^{2}$:    $x^{4} - 20 x^{2} + 64 = 1 \left(x^{2} - 16\right) \left(x^{2} - 4\right)$.

$$\color{red}{\left(x^{4} - 20 x^{2} + 64\right)} = \color{red}{1 \left(x^{2} - 16\right) \left(x^{2} - 4\right)}$$

Apply the difference of squares formula $\alpha^{2} - \beta^{2} = \left(\alpha - \beta\right) \left(\alpha + \beta\right)$ with $\alpha = x$ and $\beta = 2$:

$$\left(x^{2} - 16\right) \color{red}{\left(x^{2} - 4\right)} = \left(x^{2} - 16\right) \color{red}{\left(x - 2\right) \left(x + 2\right)}$$

Apply the difference of squares formula $\alpha^{2} - \beta^{2} = \left(\alpha - \beta\right) \left(\alpha + \beta\right)$ with $\alpha = x$ and $\beta = 4$:

$$\left(x - 2\right) \left(x + 2\right) \color{red}{\left(x^{2} - 16\right)} = \left(x - 2\right) \left(x + 2\right) \color{red}{\left(x - 4\right) \left(x + 4\right)}$$

Thus, $x^{4} - 20 x^{2} + 64=\left(x - 4\right) \left(x - 2\right) \left(x + 2\right) \left(x + 4\right)$.

Answer: $x^{4} - 20 x^{2} + 64=\left(x - 4\right) \left(x - 2\right) \left(x + 2\right) \left(x + 4\right)$.