Solve -12x^4+32x^3-15x^2-3x+2

$$- 12 x ^ { 4 } + 32 x ^ { 3 } - 15 x ^ { 2 } - 3 x + 2$$

Factor

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

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

$$\left(3x+1\right)\left(-4x^{3}+12x^{2}-9x+2\right)$$

Consider -4x^{3}+12x^{2}-9x+2. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 2 and q divides the leading coefficient -4. One such root is \frac{1}{2}. Factor the polynomial by dividing it by 2x-1.

$$\left(2x-1\right)\left(-2x^{2}+5x-2\right)$$

Consider -2x^{2}+5x-2. Factor the expression by grouping. First, the expression needs to be rewritten as -2x^{2}+ax+bx-2. To find a and b, set up a system to be solved.

$$a+b=5 ab=-2\left(-2\right)=4$$

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 4.

$$1,4 2,2$$

Calculate the sum for each pair.

$$1+4=5 2+2=4$$

The solution is the pair that gives sum 5.

$$a=4 b=1$$

Rewrite -2x^{2}+5x-2 as \left(-2x^{2}+4x\right)+\left(x-2\right).

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

Factor out 2x in the first and -1 in the second group.

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

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

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

Rewrite the complete factored expression.

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

Evaluate

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