Find roots of polynomial $$ p(x) = s^3+30s^2+316s+1160 $$
Answer
The roots of polynomial $ p(s) $ are:
$$ \begin{aligned}s_1 &= -10\\[1 em]s_2 &= -10+4i\\[1 em]s_3 &= -10-4i \end{aligned} $$Explanation
Step 1:
Use rational root test to find out that the $ \color{blue}{ s = -10 } $ is a root of polynomial $ s^3+30s^2+316s+1160 $.
The Rational Root Theorem tells us that if the polynomial has a rational zero then it must be a fraction $ \dfrac{ \color{blue}{p}}{ \color{red}{q} } $, where $ p $ is a factor of the constant term and $ q $ is a factor of the leading coefficient.
The constant term is $ \color{blue}{ 1160 } $, with a single factor of 1, 2, 4, 5, 8, 10, 20, 29, 40, 58, 116, 145, 232, 290, 580 and 1160.
The leading coefficient is $ \color{red}{ 1 }$, with a single factor of 1.
The POSSIBLE zeroes are:
$$ \begin{aligned} \dfrac{\color{blue}{p}}{\color{red}{q}} = & \dfrac{ \text{ factors of 1160 }}{\text{ factors of 1 }} = \pm \dfrac{\text{ ( 1, 2, 4, 5, 8, 10, 20, 29, 40, 58, 116, 145, 232, 290, 580, 1160 ) }}{\text{ ( 1 ) }} = \\[1 em] = & \pm \frac{ 1}{ 1} \pm \frac{ 2}{ 1} \pm \frac{ 4}{ 1} \pm \frac{ 5}{ 1} \pm \frac{ 8}{ 1} \pm \frac{ 10}{ 1} \pm \frac{ 20}{ 1} \pm \frac{ 29}{ 1} \pm \frac{ 40}{ 1} \pm \frac{ 58}{ 1} \pm \frac{ 116}{ 1} \pm \frac{ 145}{ 1} \pm \frac{ 232}{ 1} \pm \frac{ 290}{ 1} \pm \frac{ 580}{ 1} \pm \frac{ 1160}{ 1} ~~ \end{aligned} $$Substitute the possible roots one by one into the polynomial to find the actual roots. Start first with the whole numbers.
We can see that $ p\left( -10 \right) = 0 $ so $ x = -10 $ is a root of a polynomial $ p(x) $.
To find remaining zeros we use Factor Theorem. This theorem states that if $ \dfrac{p}{q} $ is root of the polynomial then the polynomial can be divided by $ \color{blue}{qx − p} $. In this example we divide polynomial $ p $ by $ \color{blue}{ s+10 }$
$$ \frac{ s^3+30s^2+316s+1160}{ s+10} = s^2+20s+116 $$Step 2:
The next rational root is $ s = -10 $
$$ \frac{ s^3+30s^2+316s+1160}{ s+10} = s^2+20s+116 $$Step 3:
The solutions of $ s^2+20s+116 = 0 $ are: $ s = -10+4i ~ \text{and} ~ s = -10-4i$.
You can use step-by-step quadratic equation solver to see a detailed explanation on how to solve this quadratic equation.