Solve 7x^4+36x^3+46x^2+x-6

$$7 x ^ { 4 } + 36 x ^ { 3 } + 46 x ^ { 2 } + x - 6$$

Factor

$$\left(x+2\right)\left(x+3\right)\left(7x^{2}+x-1\right)$$

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 -6 and q divides the leading coefficient 7. One such root is -3. Factor the polynomial by dividing it by x+3.

$$\left(x+3\right)\left(7x^{3}+15x^{2}+x-2\right)$$

Consider 7x^{3}+15x^{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 7. One such root is -2. Factor the polynomial by dividing it by x+2.

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

Rewrite the complete factored expression. Polynomial 7x^{2}+x-1 is not factored since it does not have any rational roots.

$$\left(7x^{2}+x-1\right)\left(x+2\right)\left(x+3\right)$$

Evaluate

$$\left(x+2\right)\left(x+3\right)\left(7x^{2}+x-1\right)$$