Solve f(x)=-x^3+15x^2-75x+125

$$f ( x ) = - x ^ { 3 } + 15 x ^ { 2 } - 75 x + 125$$

Factor

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

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

$$\left(x-5\right)\left(-x^{2}+10x-25\right)$$

Consider -x^{2}+10x-25. Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx-25. To find a and b, set up a system to be solved.

$$a+b=10 ab=-\left(-25\right)=25$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 25.

$$1,25 5,5$$

Calculate the sum for each pair.

$$1+25=26 5+5=10$$

The solution is the pair that gives sum 10.

$$a=5 b=5$$

Rewrite -x^{2}+10x-25 as \left(-x^{2}+5x\right)+\left(5x-25\right).

$$\left(-x^{2}+5x\right)+\left(5x-25\right)$$

Factor out -x in the first and 5 in the second group.

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

Factor out common term x-5 by using distributive property.

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

Rewrite the complete factored expression.

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

Evaluate

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