Find roots of polynomial $$ p(x) = s^5-15s^4+85s^3-225s^2+274s-120 $$
Answer
The roots of polynomial $ p(s) $ are:
$$ \begin{aligned}s_1 &= 1\\[1 em]s_2 &= 2\\[1 em]s_3 &= 3\\[1 em]s_4 &= 4\\[1 em]s_5 &= 5 \end{aligned} $$Explanation
Step 1:
Use rational root test to find out that the $ \color{blue}{ s = 1 } $ is a root of polynomial $ s^5-15s^4+85s^3-225s^2+274s-120 $.
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}{ 120 } $, with a single factor of 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60 and 120.
The leading coefficient is $ \color{red}{ 1 }$, with a single factor of 1.
The POSSIBLE zeroes are:
$$ \begin{aligned} \dfrac{\color{blue}{p}}{\color{red}{q}} = & \dfrac{ \text{ factors of 120 }}{\text{ factors of 1 }} = \pm \dfrac{\text{ ( 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120 ) }}{\text{ ( 1 ) }} = \\[1 em] = & \pm \frac{ 1}{ 1} \pm \frac{ 2}{ 1} \pm \frac{ 3}{ 1} \pm \frac{ 4}{ 1} \pm \frac{ 5}{ 1} \pm \frac{ 6}{ 1} \pm \frac{ 8}{ 1} \pm \frac{ 10}{ 1} \pm \frac{ 12}{ 1} \pm \frac{ 15}{ 1} \pm \frac{ 20}{ 1} \pm \frac{ 24}{ 1} \pm \frac{ 30}{ 1} \pm \frac{ 40}{ 1} \pm \frac{ 60}{ 1} \pm \frac{ 120}{ 1} ~~ \end{aligned} $$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 }$
$$ \frac{ s^5-15s^4+85s^3-225s^2+274s-120}{ s-1} = s^4-14s^3+71s^2-154s+120 $$Step 2:
The next rational root is $ s = 1 $
$$ \frac{ s^5-15s^4+85s^3-225s^2+274s-120}{ s-1} = s^4-14s^3+71s^2-154s+120 $$Step 3:
The next rational root is $ s = 2 $
$$ \frac{ s^4-14s^3+71s^2-154s+120}{ s-2} = s^3-12s^2+47s-60 $$Step 4:
The next rational root is $ s = 3 $
$$ \frac{ s^3-12s^2+47s-60}{ s-3} = s^2-9s+20 $$Step 5:
The next rational root is $ s = 4 $
$$ \frac{ s^2-9s+20}{ s-4} = s-5 $$Step 6:
To find the last zero, solve equation $ s-5 = 0 $
$$ \begin{aligned} s-5 & = 0 \\[1 em] s & = 5 \end{aligned} $$