Solve 2x^4+9x^3-9x^2-46x+24

$$2 x ^ { 4 } + 9 x ^ { 3 } - 9 x ^ { 2 } - 46 x + 24$$

Factor

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

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 24 and q divides the leading coefficient 2. One such root is -4. Factor the polynomial by dividing it by x+4.

$$\left(x+4\right)\left(2x^{3}+x^{2}-13x+6\right)$$

Consider 2x^{3}+x^{2}-13x+6. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 6 and q divides the leading coefficient 2. One such root is -3. Factor the polynomial by dividing it by x+3.

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

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.

$$a+b=-5 ab=2\times 2=4$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. 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.

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

Evaluate

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