Find roots of polynomial $$ p(x) = -\frac{27}{2}x^4+\frac{27}{2}x^3+\frac{1}{2}x-\frac{1}{2} $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= 1\\[1 em]x_2 &= \frac{ 1 }{ 3 }\\[1 em]x_3 &= -\frac{ 1 }{ 6 }+\frac{\sqrt{ 3 }}{ 6 }i\\[1 em]x_4 &= -\frac{ 1 }{ 6 }- \frac{\sqrt{ 3 }}{ 6 }i \end{aligned} $$Explanation
Step 1:
Get rid of fractions by multipling equation by $ \color{blue}{ 2 } $.
$$ \begin{aligned} -\frac{27}{2}x^4+\frac{27}{2}x^3+\frac{1}{2}x-\frac{1}{2} & = 0 ~~~ / \cdot \color{blue}{ 2 } \\[1 em] -27x^4+27x^3+x-1 & = 0 \end{aligned} $$Step 2:
Use rational root test to find out that the $ \color{blue}{ x = 1 } $ is a root of polynomial $ -27x^4+27x^3+x-1 $.
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}{ 1 } $, with factors of 1.
The leading coefficient is $ \color{red}{ 27 }$, with factors of 1, 3, 9 and 27.
The POSSIBLE zeroes are:
$$ \begin{aligned} \dfrac{\color{blue}{p}}{\color{red}{q}} = & \dfrac{ \text{ factors of 1 }}{\text{ factors of 27 }} = \pm \dfrac{\text{ ( 1 ) }}{\text{ ( 1, 3, 9, 27 ) }} = \\[1 em] = & \pm \frac{ 1}{ 1} ~~ \pm \frac{ 1}{ 3} ~~ \pm \frac{ 1}{ 9} ~~ \pm \frac{ 1}{ 27}\\[ 1 em] \end{aligned} $$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}{ x-1 }$
$$ \frac{ -27x^4+27x^3+x-1}{ x-1} = -27x^3+1 $$Step 3:
The next rational root is $ x = 1 $
$$ \frac{ -27x^4+27x^3+x-1}{ x-1} = -27x^3+1 $$Step 4:
The next rational root is $ x = \dfrac{ 1 }{ 3 } $
$$ \frac{ -27x^3+1}{ 3x-1} = -9x^2-3x-1 $$Step 5:
The solutions of $ -9x^2-3x-1 = 0 $ are: $ x = -\dfrac{ 1 }{ 6 }+\dfrac{\sqrt{ 3 }}{ 6 }i ~ \text{and} ~ x = -\dfrac{ 1 }{ 6 }-\dfrac{\sqrt{ 3 }}{ 6 }i$.
You can use step-by-step quadratic equation solver to see a detailed explanation on how to solve this quadratic equation.