Solve 4x^3-31x+15

$$4 { x }^{ 3 } -31x+15$$

Factor

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

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.

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

Consider 2x^{2}+5x-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.

$$a+b=5 ab=2\left(-3\right)=-6$$

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 -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 5.

$$a=-1 b=6$$

Rewrite 2x^{2}+5x-3 as \left(2x^{2}-x\right)+\left(6x-3\right).

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

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

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

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

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

Rewrite the complete factored expression.

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

Evaluate

$$4x^{3}-31x+15$$