Find roots of polynomial $$ p(x) = x^3-2028x+19773 $$

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

$$ \begin{aligned}x_1 &= 39\\[1 em]x_2 &= 10.2867\\[1 em]x_3 &= -49.2867 \end{aligned} $$

Step 1:

Use rational root test to find out that the $ \color{blue}{ x = 39 } $ is a root of polynomial $ x^3-2028x+19773 $.

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}{ 19773 } $, with a single factor of 1, 3, 9, 13, 39, 117, 169, 507, 1521, 2197, 6591 and 19773.

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

The POSSIBLE zeroes are:

$$ \begin{aligned} \dfrac{\color{blue}{p}}{\color{red}{q}} = & \dfrac{ \text{ factors of 19773 }}{\text{ factors of 1 }} = \pm \dfrac{\text{ ( 1, 3, 9, 13, 39, 117, 169, 507, 1521, 2197, 6591, 19773 ) }}{\text{ ( 1 ) }} = \\[1 em] = & \pm \frac{ 1}{ 1} \pm \frac{ 3}{ 1} \pm \frac{ 9}{ 1} \pm \frac{ 13}{ 1} \pm \frac{ 39}{ 1} \pm \frac{ 117}{ 1} \pm \frac{ 169}{ 1} \pm \frac{ 507}{ 1} \pm \frac{ 1521}{ 1} \pm \frac{ 2197}{ 1} \pm \frac{ 6591}{ 1} \pm \frac{ 19773}{ 1} ~~ \end{aligned} $$

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( 39 \right) = 0 $ so $ x = 39 $ 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-39 }$

$$ \frac{ x^3-2028x+19773}{ x-39} = x^2+39x-507 $$

Step 2:

The next rational root is $ x = 39 $

$$ \frac{ x^3-2028x+19773}{ x-39} = x^2+39x-507 $$

Step 3:

The solutions of $ x^2+39x-507 = 0 $ are: $ x = -\dfrac{ 39 }{ 2 }-\dfrac{ 13 \sqrt{ 21}}{ 2 } ~ \text{and} ~ x = -\dfrac{ 39 }{ 2 }+\dfrac{ 13 \sqrt{ 21}}{ 2 }$.

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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