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Factoring Polynomials Calculator
Factor polynomials step by step
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 $$$\frac{n^{3}}{3} + \frac{n^{2}}{2} + \frac{n}{6}$$$.
Factor the common term:
$${\color{red}{\left(\frac{n^{3}}{3} + \frac{n^{2}}{2} + \frac{n}{6}\right)}} = {\color{red}{\frac{n}{6} \left(2 n^{2} + 3 n + 1\right)}}$$
To factor the quadratic function $$$2 n^{2} + 3 n + 1$$$, we should solve the corresponding quadratic equation $$$2 n^{2} + 3 n + 1=0$$$.
Indeed, if $$$n_1$$$ and $$$n_2$$$ are the roots of the quadratic equation $$$an^2+bn+c=0$$$, then $$$an^2+bn+c=a(n-n_1)(n-n_2)$$$.
Solve the quadratic equation $$$2 n^{2} + 3 n + 1=0$$$.
The roots are $$$n_{1} = - \frac{1}{2}$$$, $$$n_{2} = -1$$$ (use the quadratic equation calculator to see the steps).
Therefore, $$$2 n^{2} + 3 n + 1 = 2 \left(n + \frac{1}{2}\right) \left(n + 1\right)$$$.
$$\frac{n {\color{red}{\left(2 n^{2} + 3 n + 1\right)}}}{6} = \frac{n {\color{red}{\left(2 \left(n + \frac{1}{2}\right) \left(n + 1\right)\right)}}}{6}$$
Simplify: $$$\frac{n \left(n + \frac{1}{2}\right) \left(n + 1\right)}{3}=\frac{n \left(n + 1\right) \left(2 n + 1\right)}{6}$$$.
Thus, $$$\frac{n^{3}}{3} + \frac{n^{2}}{2} + \frac{n}{6}=\frac{n \left(n + 1\right) \left(2 n + 1\right)}{6}$$$.
Answer: $$$\frac{n^{3}}{3} + \frac{n^{2}}{2} + \frac{n}{6}=\frac{n \left(n + 1\right) \left(2 n + 1\right)}{6}$$$.