Solve f(x)=-1/2x^3-3/2x^2+2

$$f ( x ) = - \frac { 1 } { 2 } x ^ { 3 } - \frac { 3 } { 2 } x ^ { 2 } + 2$$

Factor

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

Solution Steps

Factor out \frac{1}{2}.

$$\frac{-x^{3}-3x^{2}+4}{2}$$

Consider -x^{3}-3x^{2}+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 -1. One such root is -2. Factor the polynomial by dividing it by x+2.

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

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

$$a+b=-1 ab=-2=-2$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. The only such pair is the system solution.

$$a=1 b=-2$$

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

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

Factor out x in the first and 2 in the second group.

$$x\left(-x+1\right)+2\left(-x+1\right)$$

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

$$\left(-x+1\right)\left(x+2\right)$$

Rewrite the complete factored expression.

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

Evaluate

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