Find roots of polynomial $$ p(x) = 9x^3-57x^2+67x-35 $$

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

$$ \begin{aligned}x_1 &= 5\\[1 em]x_2 &= \frac{ 2 }{ 3 }+\frac{\sqrt{ 3 }}{ 3 }i\\[1 em]x_3 &= \frac{ 2 }{ 3 }- \frac{\sqrt{ 3 }}{ 3 }i \end{aligned} $$

Step 1:

Use rational root test to find out that the $ \color{blue}{ x = 5 } $ is a root of polynomial $ 9x^3-57x^2+67x-35 $.

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}{ 35 } $, with factors of 1, 5, 7 and 35.

The leading coefficient is $ \color{red}{ 9 }$, with factors of 1, 3 and 9.

The POSSIBLE zeroes are:

$$ \begin{aligned} \dfrac{\color{blue}{p}}{\color{red}{q}} = & \dfrac{ \text{ factors of 35 }}{\text{ factors of 9 }} = \pm \dfrac{\text{ ( 1, 5, 7, 35 ) }}{\text{ ( 1, 3, 9 ) }} = \\[1 em] = & \pm \frac{ 1}{ 1} \pm \frac{ 5}{ 1} \pm \frac{ 7}{ 1} \pm \frac{ 35}{ 1} ~~ \pm \frac{ 1}{ 3} \pm \frac{ 5}{ 3} \pm \frac{ 7}{ 3} \pm \frac{ 35}{ 3} ~~ \pm \frac{ 1}{ 9} \pm \frac{ 5}{ 9} \pm \frac{ 7}{ 9} \pm \frac{ 35}{ 9} ~~ \end{aligned} $$

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 }$

$$ \frac{ 9x^3-57x^2+67x-35}{ x-5} = 9x^2-12x+7 $$

Step 2:

The next rational root is $ x = 5 $

$$ \frac{ 9x^3-57x^2+67x-35}{ x-5} = 9x^2-12x+7 $$

Step 3:

The solutions of $ 9x^2-12x+7 = 0 $ are: $ x = \dfrac{ 2 }{ 3 }+\dfrac{\sqrt{ 3 }}{ 3 }i ~ \text{and} ~ x = \dfrac{ 2 }{ 3 }-\dfrac{\sqrt{ 3 }}{ 3 }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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