Solve P(x)=4x^3+20x^2+33x+18

$$P ( x ) = 4 x ^ { 3 } + 20 x ^ { 2 } + 33 x + 18$$

Factor

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

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 18 and q divides the leading coefficient 4. One such root is -\frac{3}{2}. Factor the polynomial by dividing it by 2x+3.

$$\left(2x+3\right)\left(2x^{2}+7x+6\right)$$

Consider 2x^{2}+7x+6. Factor the expression by grouping. First, the expression needs to be rewritten as 2x^{2}+ax+bx+6. To find a and b, set up a system to be solved.

$$a+b=7 ab=2\times 6=12$$

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 12.

$$1,12 2,6 3,4$$

Calculate the sum for each pair.

$$1+12=13 2+6=8 3+4=7$$

The solution is the pair that gives sum 7.

$$a=3 b=4$$

Rewrite 2x^{2}+7x+6 as \left(2x^{2}+3x\right)+\left(4x+6\right).

$$\left(2x^{2}+3x\right)+\left(4x+6\right)$$

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

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

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

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

Rewrite the complete factored expression.

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

Evaluate

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