因数分解計算機
式をステップバイステップで因数分解
この計算機は、手順を示しながら、任意の式（多項式、二項式、三項式、二次式、有理式、無理式、指数式、三角関数の式、またはそれらの混合）を因数分解しようとします。これを行うために、まずいくつかの代入を行って式を多項式に変換し、その後、次の手法を用います：単項式（共通因数）でくくる、二次式の因数分解、グルーピング（組分け）および再グルーピング、和・差の平方、和・差の立方、平方の差、立方の和・差、有理根の定理。
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: $$$x^{4} - 20 x^{2} + 64=\left(x - 4\right) \left(x - 2\right) \left(x + 2\right) \left(x + 4\right)$$$.