Find roots of polynomial $$ p(x) = x^9-4x^7-x^6+4x^4 $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= 1\\[1 em]x_3 &= 2\\[1 em]x_4 &= -2\\[1 em]x_5 &= -\frac{ 1 }{ 2 }+\frac{\sqrt{ 3 }}{ 2 }i\\[1 em]x_6 &= -\frac{ 1 }{ 2 }- \frac{\sqrt{ 3 }}{ 2 }i \end{aligned} $$Explanation
Step 1:
Factor out $ \color{blue}{ x^4 }$ from $ x^9-4x^7-x^6+4x^4 $ and solve two separate equations:
$$ \begin{aligned} x^9-4x^7-x^6+4x^4 & = 0\\[1 em] \color{blue}{ x^4 }\cdot ( x^5-4x^3-x^2+4 ) & = 0 \\[1 em] \color{blue}{ x^4 = 0} ~~ \text{or} ~~ x^5-4x^3-x^2+4 & = 0 \end{aligned} $$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 = 1 } $ is a root of polynomial $ x^5-4x^3-x^2+4 $.
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}{ 4 } $, with a single factor of 1, 2 and 4.
The leading coefficient is $ \color{red}{ 1 }$, with a single factor of 1.
The POSSIBLE zeroes are:
$$ \begin{aligned} \dfrac{\color{blue}{p}}{\color{red}{q}} = & \dfrac{ \text{ factors of 4 }}{\text{ factors of 1 }} = \pm \dfrac{\text{ ( 1, 2, 4 ) }}{\text{ ( 1 ) }} = \\[1 em] = & \pm \frac{ 1}{ 1} \pm \frac{ 2}{ 1} \pm \frac{ 4}{ 1} ~~ \end{aligned} $$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( 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 }$
$$ \frac{ x^5-4x^3-x^2+4}{ x-1} = x^4+x^3-3x^2-4x-4 $$Step 3:
The next rational root is $ x = 1 $
$$ \frac{ x^5-4x^3-x^2+4}{ x-1} = x^4+x^3-3x^2-4x-4 $$Step 4:
The next rational root is $ x = 2 $
$$ \frac{ x^4+x^3-3x^2-4x-4}{ x-2} = x^3+3x^2+3x+2 $$Step 5:
The next rational root is $ x = -2 $
$$ \frac{ x^3+3x^2+3x+2}{ x+2} = x^2+x+1 $$Step 6:
The solutions of $ x^2+x+1 = 0 $ are: $ x = -\dfrac{ 1 }{ 2 }+\dfrac{\sqrt{ 3 }}{ 2 }i ~ \text{and} ~ x = -\dfrac{ 1 }{ 2 }-\dfrac{\sqrt{ 3 }}{ 2 }i$.
You can use step-by-step quadratic equation solver to see a detailed explanation on how to solve this quadratic equation.