Solve x^4-3x^3-9x^2+23x-12

$$x ^ { 4 } - 3 x ^ { 3 } - 9 x ^ { 2 } + 23 x - 12$$

Factor

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

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 1. One such root is 4. Factor the polynomial by dividing it by x-4.

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

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

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

Consider x^{2}-2x+1. Use the perfect square formula, a^{2}-2ab+b^{2}=\left(a-b\right)^{2}, where a=x and b=1.

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

Rewrite the complete factored expression.

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

Evaluate

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