Solve -4x^3-13x^2+9

$$- 4 x ^ { 3 } - 13 x ^ { 2 } + 9$$

Factor

$$\left(-x-1\right)\left(4x-3\right)\left(x+3\right)$$

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 9 and q divides the leading coefficient -4. One such root is \frac{3}{4}. Factor the polynomial by dividing it by 4x-3.

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

Consider -x^{2}-4x-3. Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx-3. To find a and b, set up a system to be solved.

$$a+b=-4 ab=-\left(-3\right)=3$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. The only such pair is the system solution.

$$a=-1 b=-3$$

Rewrite -x^{2}-4x-3 as \left(-x^{2}-x\right)+\left(-3x-3\right).

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

Factor out x in the first and 3 in the second group.

$$x\left(-x-1\right)+3\left(-x-1\right)$$

Factor out common term -x-1 by using distributive property.

$$\left(-x-1\right)\left(x+3\right)$$

Rewrite the complete factored expression.

$$\left(4x-3\right)\left(-x-1\right)\left(x+3\right)$$

Evaluate

$$9-13x^{2}-4x^{3}$$