Solve x^3+12x^2+44x+48

$$x ^ { 3 } + 12 x ^ { 2 } + 44 x + 48$$

Factor

$$\left(x+2\right)\left(x+4\right)\left(x+6\right)$$

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 48 and q divides the leading coefficient 1. One such root is -6. Factor the polynomial by dividing it by x+6.

$$\left(x+6\right)\left(x^{2}+6x+8\right)$$

Consider x^{2}+6x+8. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+8. To find a and b, set up a system to be solved.

$$a+b=6 ab=1\times 8=8$$

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 8.

$$1,8 2,4$$

Calculate the sum for each pair.

$$1+8=9 2+4=6$$

The solution is the pair that gives sum 6.

$$a=2 b=4$$

Rewrite x^{2}+6x+8 as \left(x^{2}+2x\right)+\left(4x+8\right).

$$\left(x^{2}+2x\right)+\left(4x+8\right)$$

Factor out x in the first and 4 in the second group.

$$x\left(x+2\right)+4\left(x+2\right)$$

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

$$\left(x+2\right)\left(x+4\right)$$

Rewrite the complete factored expression.

$$\left(x+2\right)\left(x+4\right)\left(x+6\right)$$

Evaluate

$$\left(x+2\right)\left(x+4\right)\left(x+6\right)$$