Solve f(x)=2x^3+x^2-25x+12

$$f ( x ) = 2 x ^ { 3 } + x ^ { 2 } - 25 x + 12$$

Factor

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

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 12 and q divides the leading coefficient 2. One such root is 3. Factor the polynomial by dividing it by x-3.

$$\left(x-3\right)\left(2x^{2}+7x-4\right)$$

Consider 2x^{2}+7x-4. Factor the expression by grouping. First, the expression needs to be rewritten as 2x^{2}+ax+bx-4. To find a and b, set up a system to be solved.

$$a+b=7 ab=2\left(-4\right)=-8$$

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 -8.

$$-1,8 -2,4$$

Calculate the sum for each pair.

$$-1+8=7 -2+4=2$$

The solution is the pair that gives sum 7.

$$a=-1 b=8$$

Rewrite 2x^{2}+7x-4 as \left(2x^{2}-x\right)+\left(8x-4\right).

$$\left(2x^{2}-x\right)+\left(8x-4\right)$$

Factor out x in the first and 4 in the second group.

$$x\left(2x-1\right)+4\left(2x-1\right)$$

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

$$\left(2x-1\right)\left(x+4\right)$$

Rewrite the complete factored expression.

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

Evaluate

$$2x^{3}+x^{2}-25x+12$$