Solve 12x^3+16x^2-5x-3

$$12 x ^ { 3 } + 16 x ^ { 2 } - 5 x - 3$$

Factor

$$\left(2x-1\right)\left(2x+3\right)\left(3x+1\right)$$

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 -3 and q divides the leading coefficient 12. One such root is -\frac{1}{3}. Factor the polynomial by dividing it by 3x+1.

$$\left(3x+1\right)\left(4x^{2}+4x-3\right)$$

Consider 4x^{2}+4x-3. Factor the expression by grouping. First, the expression needs to be rewritten as 4x^{2}+ax+bx-3. To find a and b, set up a system to be solved.

$$a+b=4 ab=4\left(-3\right)=-12$$

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 -12.

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

Calculate the sum for each pair.

$$-1+12=11 -2+6=4 -3+4=1$$

The solution is the pair that gives sum 4.

$$a=-2 b=6$$

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

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

Factor out 2x in the first and 3 in the second group.

$$2x\left(2x-1\right)+3\left(2x-1\right)$$

Factor out common term 2x-1 by using distributive property.

$$\left(2x-1\right)\left(2x+3\right)$$

Rewrite the complete factored expression.

$$\left(2x-1\right)\left(3x+1\right)\left(2x+3\right)$$

Evaluate

$$\left(2x-1\right)\left(2x+3\right)\left(3x+1\right)$$