Solve x^4+3x^3-15x^2-45x-28

$$x ^ { 4 } + 3 x ^ { 3 } - 15 x ^ { 2 } - 45 x - 28$$

Factor

$$\left(x-4\right)\left(x+1\right)\left(x^{2}+6x+7\right)$$

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 -28 and q divides the leading coefficient 1. One such root is 4. Factor the polynomial by dividing it by x-4.

$$\left(x-4\right)\left(x^{3}+7x^{2}+13x+7\right)$$

Consider x^{3}+7x^{2}+13x+7. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 7 and q divides the leading coefficient 1. One such root is -1. Factor the polynomial by dividing it by x+1.

$$\left(x+1\right)\left(x^{2}+6x+7\right)$$

Rewrite the complete factored expression. Polynomial x^{2}+6x+7 is not factored since it does not have any rational roots.

$$\left(x-4\right)\left(x+1\right)\left(x^{2}+6x+7\right)$$

Evaluate

$$\left(x-4\right)\left(x+1\right)\left(x^{2}+6x+7\right)$$