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Expresiones factoriales paso a paso

La calculadora intentará factorizar cualquier expresión (polinomial, binomial, trinomio, cuadrática, racional, irracional, exponencial, trigonométrica o una combinación de ellas), mostrando los pasos. Para ello, primero se aplican algunas sustituciones para convertir la expresión en un polinomio, y luego se utilizan las siguientes técnicas: factorización de monomios (factor común), factorización de cuadráticas, agrupación y reagrupación, cuadrado de suma/diferencia, cubo de suma/diferencia , diferencia de cuadrados, suma/diferencia de cubos y el teorema de los ceros racionales.

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