Solve 5x^4+46x^3+78x^2+5x-14

$$5 x ^ { 4 } + 46 x ^ { 3 } + 78 x ^ { 2 } + 5 x - 14$$

Factor

$$\left(x+2\right)\left(x+7\right)\left(5x^{2}+x-1\right)$$

Solution Steps

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -14 and q divides the leading coefficient 5. One such root is -7. Factor the polynomial by dividing it by x+7.

$$\left(x+7\right)\left(5x^{3}+11x^{2}+x-2\right)$$

Consider 5x^{3}+11x^{2}+x-2. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -2 and q divides the leading coefficient 5. One such root is -2. Factor the polynomial by dividing it by x+2.

$$\left(x+2\right)\left(5x^{2}+x-1\right)$$

Rewrite the complete factored expression. Polynomial 5x^{2}+x-1 is not factored since it does not have any rational roots.

$$\left(5x^{2}+x-1\right)\left(x+2\right)\left(x+7\right)$$

Evaluate

$$\left(x+2\right)\left(x+7\right)\left(5x^{2}+x-1\right)$$