Solve f(x)=-x^3+3x^2+13x-15

$$f ( x ) = - x ^ { 3 } + 3 x ^ { 2 } + 13 x - 15$$

Factor

$$\left(1-x\right)\left(x-5\right)\left(x+3\right)$$

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 -15 and q divides the leading coefficient -1. One such root is 5. Factor the polynomial by dividing it by x-5.

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

Consider -x^{2}-2x+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.

$$a+b=-2 ab=-3=-3$$

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.

$$a=1 b=-3$$

Rewrite -x^{2}-2x+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.

$$\left(x-5\right)\left(-x+1\right)\left(x+3\right)$$

Evaluate

$$\left(1-x\right)\left(x-5\right)\left(x+3\right)$$