Solve g(x)=x^3+8x^2+17x+10

$$g ( x ) = x ^ { 3 } + 8 x ^ { 2 } + 17 x + 10$$

Factor

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

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 10 and q divides the leading coefficient 1. One such root is -5. Factor the polynomial by dividing it by x+5.

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

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

$$a+b=3 ab=1\times 2=2$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. The only such pair is the system solution.

$$a=1 b=2$$

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

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

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

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

Factor out common term x+1 by using distributive property.

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

Rewrite the complete factored expression.

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

Evaluate

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