Solve 2n^4+4n^3+2n^2-3528

$$2 n ^ { 4 } + 4 n ^ { 3 } + 2 n ^ { 2 } - 3528$$

Factor

$$2\left(n-6\right)\left(n+7\right)\left(n^{2}+n+42\right)$$

Solution Steps

Factor out 2.

$$2\left(n^{4}+2n^{3}+n^{2}-1764\right)$$

Consider n^{4}+2n^{3}+n^{2}-1764. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -1764 and q divides the leading coefficient 1. One such root is -7. Factor the polynomial by dividing it by n+7.

$$\left(n+7\right)\left(n^{3}-5n^{2}+36n-252\right)$$

Consider n^{3}-5n^{2}+36n-252. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -252 and q divides the leading coefficient 1. One such root is 6. Factor the polynomial by dividing it by n-6.

$$\left(n-6\right)\left(n^{2}+n+42\right)$$

Rewrite the complete factored expression. Polynomial n^{2}+n+42 is not factored since it does not have any rational roots.

$$2\left(n+7\right)\left(n-6\right)\left(n^{2}+n+42\right)$$

Evaluate

$$2\left(n-6\right)\left(n+7\right)\left(n^{2}+n+42\right)$$