Solve -x^3+15x^2-66x+80=

$$- x ^ { 3 } + 15 x ^ { 2 } - 66 x + 80 =$$

Factor

$$\left(2-x\right)\left(x-8\right)\left(x-5\right)$$

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 80 and q divides the leading coefficient -1. One such root is 8. Factor the polynomial by dividing it by x-8.

$$\left(x-8\right)\left(-x^{2}+7x-10\right)$$

Consider -x^{2}+7x-10. Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx-10. To find a and b, set up a system to be solved.

$$a+b=7 ab=-\left(-10\right)=10$$

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 10.

$$1,10 2,5$$

Calculate the sum for each pair.

$$1+10=11 2+5=7$$

The solution is the pair that gives sum 7.

$$a=5 b=2$$

Rewrite -x^{2}+7x-10 as \left(-x^{2}+5x\right)+\left(2x-10\right).

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

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

$$-x\left(x-5\right)+2\left(x-5\right)$$

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

$$\left(x-5\right)\left(-x+2\right)$$

Rewrite the complete factored expression.

$$\left(x-8\right)\left(x-5\right)\left(-x+2\right)$$

Evaluate

$$\left(2-x\right)\left(x-8\right)\left(x-5\right)$$