Find roots of polynomial $$ p(x) = x^5+20x^4+120x^3+220x^2+36x+72 $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= -6\\[1 em]x_2 &= -3.3762\\[1 em]x_3 &= -10.6428\\[1 em]x_4 &= 0.0095+0.5778i\\[1 em]x_5 &= 0.0095-0.5778i \end{aligned} $$Explanation
Step 1:
Use rational root test to find out that the $ \color{blue}{ x = -6 } $ is a root of polynomial $ x^5+20x^4+120x^3+220x^2+36x+72 $.
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}{ 72 } $, with a single factor of 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72.
The leading coefficient is $ \color{red}{ 1 }$, with a single factor of 1.
The POSSIBLE zeroes are:
$$ \begin{aligned} \dfrac{\color{blue}{p}}{\color{red}{q}} = & \dfrac{ \text{ factors of 72 }}{\text{ factors of 1 }} = \pm \dfrac{\text{ ( 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 ) }}{\text{ ( 1 ) }} = \\[1 em] = & \pm \frac{ 1}{ 1} \pm \frac{ 2}{ 1} \pm \frac{ 3}{ 1} \pm \frac{ 4}{ 1} \pm \frac{ 6}{ 1} \pm \frac{ 8}{ 1} \pm \frac{ 9}{ 1} \pm \frac{ 12}{ 1} \pm \frac{ 18}{ 1} \pm \frac{ 24}{ 1} \pm \frac{ 36}{ 1} \pm \frac{ 72}{ 1} ~~ \end{aligned} $$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( -6 \right) = 0 $ so $ x = -6 $ 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}{ x+6 }$
$$ \frac{ x^5+20x^4+120x^3+220x^2+36x+72}{ x+6} = x^4+14x^3+36x^2+4x+12 $$Step 2:
The next rational root is $ x = -6 $
$$ \frac{ x^5+20x^4+120x^3+220x^2+36x+72}{ x+6} = x^4+14x^3+36x^2+4x+12 $$Step 3:
Polynomial $ x^4+14x^3+36x^2+4x+12 $ has no rational roots that can be found using Rational Root Test, so the roots were found using quartic formulas.