Find roots of polynomial $$ p(x) = x^3-6x+9 $$

The roots of polynomial $ p(x) $ are:

$$ \begin{aligned}x_1 &= -3\\[1 em]x_2 &= \frac{ 3 }{ 2 }+\frac{\sqrt{ 3 }}{ 2 }i\\[1 em]x_3 &= \frac{ 3 }{ 2 }- \frac{\sqrt{ 3 }}{ 2 }i \end{aligned} $$

Step 1:

Use rational root test to find out that the $ \color{blue}{ x = -3 } $ is a root of polynomial $ x^3-6x+9 $.

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}{ 9 } $, with a single factor of 1, 3 and 9.

The leading coefficient is $ \color{red}{ 1 }$, with a single factor of 1.

The POSSIBLE zeroes are:

$$ \begin{aligned} \dfrac{\color{blue}{p}}{\color{red}{q}} = & \dfrac{ \text{ factors of 9 }}{\text{ factors of 1 }} = \pm \dfrac{\text{ ( 1, 3, 9 ) }}{\text{ ( 1 ) }} = \\[1 em] = & \pm \frac{ 1}{ 1} \pm \frac{ 3}{ 1} \pm \frac{ 9}{ 1} ~~ \end{aligned} $$

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 }$

$$ \frac{ x^3-6x+9}{ x+3} = x^2-3x+3 $$

Step 2:

The next rational root is $ x = -3 $

$$ \frac{ x^3-6x+9}{ x+3} = x^2-3x+3 $$

Step 3:

The solutions of $ x^2-3x+3 = 0 $ are: $ x = \dfrac{ 3 }{ 2 }+\dfrac{\sqrt{ 3 }}{ 2 }i ~ \text{and} ~ x = \dfrac{ 3 }{ 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.

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