Find roots of polynomial $$ p(x) = 2x^5+14x^4-12x^3-144x^2 $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= 3\\[1 em]x_3 &= -4\\[1 em]x_4 &= -6 \end{aligned} $$Explanation
Step 1:
Factor out $ \color{blue}{ 2x^2 }$ from $ 2x^5+14x^4-12x^3-144x^2 $ and solve two separate equations:
$$ \begin{aligned} 2x^5+14x^4-12x^3-144x^2 & = 0\\[1 em] \color{blue}{ 2x^2 }\cdot ( x^3+7x^2-6x-72 ) & = 0 \\[1 em] \color{blue}{ 2x^2 = 0} ~~ \text{or} ~~ x^3+7x^2-6x-72 & = 0 \end{aligned} $$One solution is $ \color{blue}{ x = 0 } $. Use second equation to find the remaining roots.
Step 2:
Use rational root test to find out that the $ \color{blue}{ x = 3 } $ is a root of polynomial $ x^3+7x^2-6x-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( 3 \right) = 0 $ so $ x = 3 $ 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-3 }$
$$ \frac{ x^3+7x^2-6x-72}{ x-3} = x^2+10x+24 $$Step 3:
The next rational root is $ x = 3 $
$$ \frac{ x^3+7x^2-6x-72}{ x-3} = x^2+10x+24 $$Step 4:
The next rational root is $ x = -4 $
$$ \frac{ x^2+10x+24}{ x+4} = x+6 $$Step 5:
To find the last zero, solve equation $ x+6 = 0 $
$$ \begin{aligned} x+6 & = 0 \\[1 em] x & = -6 \end{aligned} $$