Solve 4x^3-27x+27

$$4 x ^ { 3 } - 27 x + 27$$

Factor

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

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 4. One such root is \frac{3}{2}. Factor the polynomial by dividing it by 2x-3.

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

Consider 2x^{2}+3x-9. Factor the expression by grouping. First, the expression needs to be rewritten as 2x^{2}+ax+bx-9. To find a and b, set up a system to be solved.

$$a+b=3 ab=2\left(-9\right)=-18$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -18.

$$-1,18 -2,9 -3,6$$

Calculate the sum for each pair.

$$-1+18=17 -2+9=7 -3+6=3$$

The solution is the pair that gives sum 3.

$$a=-3 b=6$$

Rewrite 2x^{2}+3x-9 as \left(2x^{2}-3x\right)+\left(6x-9\right).

$$\left(2x^{2}-3x\right)+\left(6x-9\right)$$

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

$$x\left(2x-3\right)+3\left(2x-3\right)$$

Factor out common term 2x-3 by using distributive property.

$$\left(2x-3\right)\left(x+3\right)$$

Rewrite the complete factored expression.

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

Evaluate

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