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

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

Factor

$$\left(2x-5\right)\left(2x-3\right)\left(x+1\right)$$

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 15 and q divides the leading coefficient 4. One such root is \frac{5}{2}. Factor the polynomial by dividing it by 2x-5.

$$\left(2x-5\right)\left(2x^{2}-x-3\right)$$

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

$$a+b=-1 ab=2\left(-3\right)=-6$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -6.

$$1,-6 2,-3$$

Calculate the sum for each pair.

$$1-6=-5 2-3=-1$$

The solution is the pair that gives sum -1.

$$a=-3 b=2$$

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

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

Factor out x in 2x^{2}-3x.

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

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

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

Rewrite the complete factored expression.

$$\left(2x-5\right)\left(2x-3\right)\left(x+1\right)$$

Evaluate

$$\left(2x-5\right)\left(2x-3\right)\left(x+1\right)$$