Solve =4x^4-9x^3-44x^2+61x+12

$$= 4 x ^ { 4 } - 9 x ^ { 3 } - 44 x ^ { 2 } + 61 x + 12$$

Factor

$$4\left(x-4\right)\left(x+3\right)\left(x-\frac{5-\sqrt{41}}{8}\right)\left(x-\frac{\sqrt{41}+5}{8}\right)$$

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 12 and q divides the leading coefficient 4. One such root is -3. Factor the polynomial by dividing it by x+3.

$$\left(x+3\right)\left(4x^{3}-21x^{2}+19x+4\right)$$

Consider 4x^{3}-21x^{2}+19x+4. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 4 and q divides the leading coefficient 4. One such root is 4. Factor the polynomial by dividing it by x-4.

$$\left(x-4\right)\left(4x^{2}-5x-1\right)$$

Rewrite the complete factored expression. Polynomial 4x^{2}-5x-1 is not factored since it does not have any rational roots.

$$\left(x-4\right)\left(4x^{2}-5x-1\right)\left(x+3\right)$$

Evaluate

$$\left(x-4\right)\left(x+3\right)\left(4x^{2}-5x-1\right)$$