Find roots of polynomial $$ p(x) = 4x^3-15x^2-31x+30 $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= 5\\[1 em]x_2 &= -2\\[1 em]x_3 &= \frac{ 3 }{ 4 } \end{aligned} $$Explanation
Step 1:
Use rational root test to find out that the $ \color{blue}{ x = 5 } $ is a root of polynomial $ 4x^3-15x^2-31x+30 $.
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}{ 30 } $, with factors of 1, 2, 3, 5, 6, 10, 15 and 30.
The leading coefficient is $ \color{red}{ 4 }$, with factors of 1, 2 and 4.
The POSSIBLE zeroes are:
$$ \begin{aligned} \dfrac{\color{blue}{p}}{\color{red}{q}} = & \dfrac{ \text{ factors of 30 }}{\text{ factors of 4 }} = \pm \dfrac{\text{ ( 1, 2, 3, 5, 6, 10, 15, 30 ) }}{\text{ ( 1, 2, 4 ) }} = \\[1 em] = & \pm \frac{ 1}{ 1} \pm \frac{ 2}{ 1} \pm \frac{ 3}{ 1} \pm \frac{ 5}{ 1} \pm \frac{ 6}{ 1} \pm \frac{ 10}{ 1} \pm \frac{ 15}{ 1} \pm \frac{ 30}{ 1} ~~ \pm \frac{ 1}{ 2} \pm \frac{ 2}{ 2} \pm \frac{ 3}{ 2} \pm \frac{ 5}{ 2} \pm \frac{ 6}{ 2} \pm \frac{ 10}{ 2} \pm \frac{ 15}{ 2} \pm \frac{ 30}{ 2} ~~ \pm \frac{ 1}{ 4} \pm \frac{ 2}{ 4} \pm \frac{ 3}{ 4} \pm \frac{ 5}{ 4} \pm \frac{ 6}{ 4} \pm \frac{ 10}{ 4} \pm \frac{ 15}{ 4} \pm \frac{ 30}{ 4} ~~ \end{aligned} $$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( 5 \right) = 0 $ so $ x = 5 $ 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-5 }$
$$ \frac{ 4x^3-15x^2-31x+30}{ x-5} = 4x^2+5x-6 $$Step 2:
The next rational root is $ x = 5 $
$$ \frac{ 4x^3-15x^2-31x+30}{ x-5} = 4x^2+5x-6 $$Step 3:
The next rational root is $ x = -2 $
$$ \frac{ 4x^2+5x-6}{ x+2} = 4x-3 $$Step 4:
To find the last zero, solve equation $ 4x-3 = 0 $
$$ \begin{aligned} 4x-3 & = 0 \\[1 em] 4 \cdot x & = 3 \\[1 em] x & = \frac{ 3 }{ 4 } \end{aligned} $$