Solve 2x^3+17x^2+46x+40

$$2 { x }^{ 3 } +17 { x }^{ 2 } +46x+40$$

Factor

$$\left(x+2\right)\left(x+4\right)\left(2x+5\right)$$

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

$$\left(x+4\right)\left(2x^{2}+9x+10\right)$$

Consider 2x^{2}+9x+10. Factor the expression by grouping. First, the expression needs to be rewritten as 2x^{2}+ax+bx+10. To find a and b, set up a system to be solved.

$$a+b=9 ab=2\times 10=20$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 20.

$$1,20 2,10 4,5$$

Calculate the sum for each pair.

$$1+20=21 2+10=12 4+5=9$$

The solution is the pair that gives sum 9.

$$a=4 b=5$$

Rewrite 2x^{2}+9x+10 as \left(2x^{2}+4x\right)+\left(5x+10\right).

$$\left(2x^{2}+4x\right)+\left(5x+10\right)$$

Factor out 2x in the first and 5 in the second group.

$$2x\left(x+2\right)+5\left(x+2\right)$$

Factor out common term x+2 by using distributive property.

$$\left(x+2\right)\left(2x+5\right)$$

Rewrite the complete factored expression.

$$\left(x+2\right)\left(x+4\right)\left(2x+5\right)$$

Evaluate

$$\left(x+2\right)\left(x+4\right)\left(2x+5\right)$$