Solve f(x)=x^4-6x^2-8x-3

$$f ( x ) = x ^ { 4 } - 6 x ^ { 2 } - 8 x - 3$$

Factor

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

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 -3 and q divides the leading coefficient 1. One such root is 3. Factor the polynomial by dividing it by x-3.

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

Consider x^{3}+3x^{2}+3x+1. Use the binomial cube formula, a^{3}+3a^{2}b+3ab^{2}+b^{3}=\left(a+b\right)^{3}, where a=x and b=1.

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

Rewrite the complete factored expression.

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

Evaluate

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