Solve 2x^3-25x^2+19x+154

$$2 { x }^{ 3 } -25 { x }^{ 2 } +19x+154$$

Factor

$$\left(x-11\right)\left(2x-7\right)\left(x+2\right)$$

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 154 and q divides the leading coefficient 2. One such root is 11. Factor the polynomial by dividing it by x-11.

$$\left(x-11\right)\left(2x^{2}-3x-14\right)$$

Consider 2x^{2}-3x-14. Factor the expression by grouping. First, the expression needs to be rewritten as 2x^{2}+ax+bx-14. To find a and b, set up a system to be solved.

$$a+b=-3 ab=2\left(-14\right)=-28$$

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 -28.

$$1,-28 2,-14 4,-7$$

Calculate the sum for each pair.

$$1-28=-27 2-14=-12 4-7=-3$$

The solution is the pair that gives sum -3.

$$a=-7 b=4$$

Rewrite 2x^{2}-3x-14 as \left(2x^{2}-7x\right)+\left(4x-14\right).

$$\left(2x^{2}-7x\right)+\left(4x-14\right)$$

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

$$x\left(2x-7\right)+2\left(2x-7\right)$$

Factor out common term 2x-7 by using distributive property.

$$\left(2x-7\right)\left(x+2\right)$$

Rewrite the complete factored expression.

$$\left(x-11\right)\left(2x-7\right)\left(x+2\right)$$

Evaluate

$$\left(x-11\right)\left(2x-7\right)\left(x+2\right)$$