Solve 27-125x^3-135x+225x^2

$$27-125 { x }^{ 3 } -135x+225 { x }^{ 2 }$$

Factor

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

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 27 and q divides the leading coefficient -125. One such root is \frac{3}{5}. Factor the polynomial by dividing it by 5x-3.

$$\left(5x-3\right)\left(-25x^{2}+30x-9\right)$$

Consider -25x^{2}+30x-9. Factor the expression by grouping. First, the expression needs to be rewritten as -25x^{2}+ax+bx-9. To find a and b, set up a system to be solved.

$$a+b=30 ab=-25\left(-9\right)=225$$

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 225.

$$1,225 3,75 5,45 9,25 15,15$$

Calculate the sum for each pair.

$$1+225=226 3+75=78 5+45=50 9+25=34 15+15=30$$

The solution is the pair that gives sum 30.

$$a=15 b=15$$

Rewrite -25x^{2}+30x-9 as \left(-25x^{2}+15x\right)+\left(15x-9\right).

$$\left(-25x^{2}+15x\right)+\left(15x-9\right)$$

Factor out -5x in the first and 3 in the second group.

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

Factor out common term 5x-3 by using distributive property.

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

Rewrite the complete factored expression.

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

Evaluate

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