Solve 12x^3+55x^2+13x

$$12 x ^ { 3 } + 55 x ^ { 2 } + 13 x$$

Factor

$$x\left(3x+13\right)\left(4x+1\right)$$

Solution Steps

Factor out x.

$$x\left(12x^{2}+55x+13\right)$$

Consider 12x^{2}+55x+13. Factor the expression by grouping. First, the expression needs to be rewritten as 12x^{2}+ax+bx+13. To find a and b, set up a system to be solved.

$$a+b=55 ab=12\times 13=156$$

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 156.

$$1,156 2,78 3,52 4,39 6,26 12,13$$

Calculate the sum for each pair.

$$1+156=157 2+78=80 3+52=55 4+39=43 6+26=32 12+13=25$$

The solution is the pair that gives sum 55.

$$a=3 b=52$$

Rewrite 12x^{2}+55x+13 as \left(12x^{2}+3x\right)+\left(52x+13\right).

$$\left(12x^{2}+3x\right)+\left(52x+13\right)$$

Factor out 3x in the first and 13 in the second group.

$$3x\left(4x+1\right)+13\left(4x+1\right)$$

Factor out common term 4x+1 by using distributive property.

$$\left(4x+1\right)\left(3x+13\right)$$

Rewrite the complete factored expression.

$$x\left(4x+1\right)\left(3x+13\right)$$

Evaluate

$$x\left(3x+13\right)\left(4x+1\right)$$