Find roots of polynomial $$ p(x) = s^6+s^5-6s^4-s^2-s+6 $$

The roots of polynomial $ p(s) $ are:

$$ \begin{aligned}s_1 &= 1\\[1 em]s_2 &= 2\\[1 em]s_3 &= -1\\[1 em]s_4 &= -3\\[1 em]s_5 &= i\\[1 em]s_6 &= -i \end{aligned} $$

Step 1:

Use rational root test to find out that the $ \color{blue}{ s = 1 } $ is a root of polynomial $ s^6+s^5-6s^4-s^2-s+6 $.

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}{ 6 } $, with a single factor of 1, 2, 3 and 6.

The leading coefficient is $ \color{red}{ 1 }$, with a single factor of 1.

The POSSIBLE zeroes are:

$$ \begin{aligned} \dfrac{\color{blue}{p}}{\color{red}{q}} = & \dfrac{ \text{ factors of 6 }}{\text{ factors of 1 }} = \pm \dfrac{\text{ ( 1, 2, 3, 6 ) }}{\text{ ( 1 ) }} = \\[1 em] = & \pm \frac{ 1}{ 1} \pm \frac{ 2}{ 1} \pm \frac{ 3}{ 1} \pm \frac{ 6}{ 1} ~~ \end{aligned} $$

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( 1 \right) = 0 $ so $ x = 1 $ 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-1 }$

$$ \frac{ s^6+s^5-6s^4-s^2-s+6}{ s-1} = s^5+2s^4-4s^3-4s^2-5s-6 $$

Step 2:

The next rational root is $ s = 1 $

$$ \frac{ s^6+s^5-6s^4-s^2-s+6}{ s-1} = s^5+2s^4-4s^3-4s^2-5s-6 $$

Step 3:

The next rational root is $ s = 2 $

$$ \frac{ s^5+2s^4-4s^3-4s^2-5s-6}{ s-2} = s^4+4s^3+4s^2+4s+3 $$

Step 4:

The next rational root is $ s = -1 $

$$ \frac{ s^4+4s^3+4s^2+4s+3}{ s+1} = s^3+3s^2+s+3 $$

Step 5:

The next rational root is $ s = -3 $

$$ \frac{ s^3+3s^2+s+3}{ s+3} = s^2+1 $$

Step 6:

The solutions of $ s^2+1 = 0 $ are: $ s = i ~ \text{and} ~ s = -i$.

You can use step-by-step quadratic equation solver to see a detailed explanation on how to solve this quadratic equation.

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