Solve r(s)=3s^3+s^2-8s+4

$$r ( s ) = 3 s ^ { 3 } + s ^ { 2 } - 8 s + 4$$

Factor

$$\left(s-1\right)\left(3s-2\right)\left(s+2\right)$$

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 4 and q divides the leading coefficient 3. One such root is \frac{2}{3}. Factor the polynomial by dividing it by 3s-2.

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

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

$$a+b=1 ab=1\left(-2\right)=-2$$

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. The only such pair is the system solution.

$$a=-1 b=2$$

Rewrite s^{2}+s-2 as \left(s^{2}-s\right)+\left(2s-2\right).

$$\left(s^{2}-s\right)+\left(2s-2\right)$$

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

$$s\left(s-1\right)+2\left(s-1\right)$$

Factor out common term s-1 by using distributive property.

$$\left(s-1\right)\left(s+2\right)$$

Rewrite the complete factored expression.

$$\left(3s-2\right)\left(s-1\right)\left(s+2\right)$$

Evaluate

$$\left(s-1\right)\left(3s-2\right)\left(s+2\right)$$