Find the binomial that completes the factorization calculator

The calculator will try to factor any polynomial (binomial, trinomial, quadratic, etc.), with steps shown. The following methods 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, the rational zeros theorem. The calculator accepts both univariate and multivariate polynomials.

Solution

Your input: factor $$$x^{2} + 4 x + 3$$$.

To factor the quadratic function $$$x^{2} + 4 x + 3$$$, we should solve the corresponding quadratic equation $$$x^{2} + 4 x + 3=0$$$.

Indeed, if $$$x_1$$$ and $$$x_2$$$ are the roots of the quadratic equation $$$ax^2+bx+c=0$$$, then $$$ax^2+bx+c=a(x-x_1)(x-x_2)$$$.

Solve the quadratic equation $$$x^{2} + 4 x + 3=0$$$.

The roots are $$$x_{1} = -1$$$, $$$x_{2} = -3$$$ (use the quadratic equation calculator to see the steps).

Therefore, $$$x^{2} + 4 x + 3 = \left(x + 1\right) \left(x + 3\right)$$$.

$$\color{red}{\left(x^{2} + 4 x + 3\right)} = \color{red}{\left(x + 1\right) \left(x + 3\right)}$$

Thus, $$$x^{2} + 4 x + 3=\left(x + 1\right) \left(x + 3\right)$$$.

Answer: $$$x^{2} + 4 x + 3=\left(x + 1\right) \left(x + 3\right)$$$.

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Factor expressions step by step

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.

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