Find roots of polynomial $$ p(x) = 5x^3-49x^2+24x+108 $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= 2\\[1 em]x_2 &= 9\\[1 em]x_3 &= -\frac{ 6 }{ 5 } \end{aligned} $$Explanation
Step 1:
Use rational root test to find out that the $ \color{blue}{ x = 2 } $ is a root of polynomial $ 5x^3-49x^2+24x+108 $.
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}{ 108 } $, with factors of 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54 and 108.
The leading coefficient is $ \color{red}{ 5 }$, with factors of 1 and 5.
The POSSIBLE zeroes are:
$$ \begin{aligned} \dfrac{\color{blue}{p}}{\color{red}{q}} = & \dfrac{ \text{ factors of 108 }}{\text{ factors of 5 }} = \pm \dfrac{\text{ ( 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108 ) }}{\text{ ( 1, 5 ) }} = \\[1 em] = & \pm \frac{ 1}{ 1} \pm \frac{ 2}{ 1} \pm \frac{ 3}{ 1} \pm \frac{ 4}{ 1} \pm \frac{ 6}{ 1} \pm \frac{ 9}{ 1} \pm \frac{ 12}{ 1} \pm \frac{ 18}{ 1} \pm \frac{ 27}{ 1} \pm \frac{ 36}{ 1} \pm \frac{ 54}{ 1} \pm \frac{ 108}{ 1} ~~ \pm \frac{ 1}{ 5} \pm \frac{ 2}{ 5} \pm \frac{ 3}{ 5} \pm \frac{ 4}{ 5} \pm \frac{ 6}{ 5} \pm \frac{ 9}{ 5} \pm \frac{ 12}{ 5} \pm \frac{ 18}{ 5} \pm \frac{ 27}{ 5} \pm \frac{ 36}{ 5} \pm \frac{ 54}{ 5} \pm \frac{ 108}{ 5} ~~ \end{aligned} $$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( 2 \right) = 0 $ so $ x = 2 $ 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-2 }$
$$ \frac{ 5x^3-49x^2+24x+108}{ x-2} = 5x^2-39x-54 $$Step 2:
The next rational root is $ x = 2 $
$$ \frac{ 5x^3-49x^2+24x+108}{ x-2} = 5x^2-39x-54 $$Step 3:
The next rational root is $ x = 9 $
$$ \frac{ 5x^2-39x-54}{ x-9} = 5x+6 $$Step 4:
To find the last zero, solve equation $ 5x+6 = 0 $
$$ \begin{aligned} 5x+6 & = 0 \\[1 em] 5 \cdot x & = -6 \\[1 em] x & = - \frac{ 6 }{ 5 } \end{aligned} $$