Find roots of polynomial $$ p(x) = x^3+22x^2+153x+330 $$

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

$$ \begin{aligned}x_1 &= -10\\[1 em]x_2 &= -4.2679\\[1 em]x_3 &= -7.7321 \end{aligned} $$

Step 1:

Use rational root test to find out that the $ \color{blue}{ x = -10 } $ is a root of polynomial $ x^3+22x^2+153x+330 $.

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}{ 330 } $, with a single factor of 1, 2, 3, 5, 6, 10, 11, 15, 22, 30, 33, 55, 66, 110, 165 and 330.

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

The POSSIBLE zeroes are:

$$ \begin{aligned} \dfrac{\color{blue}{p}}{\color{red}{q}} = & \dfrac{ \text{ factors of 330 }}{\text{ factors of 1 }} = \pm \dfrac{\text{ ( 1, 2, 3, 5, 6, 10, 11, 15, 22, 30, 33, 55, 66, 110, 165, 330 ) }}{\text{ ( 1 ) }} = \\[1 em] = & \pm \frac{ 1}{ 1} \pm \frac{ 2}{ 1} \pm \frac{ 3}{ 1} \pm \frac{ 5}{ 1} \pm \frac{ 6}{ 1} \pm \frac{ 10}{ 1} \pm \frac{ 11}{ 1} \pm \frac{ 15}{ 1} \pm \frac{ 22}{ 1} \pm \frac{ 30}{ 1} \pm \frac{ 33}{ 1} \pm \frac{ 55}{ 1} \pm \frac{ 66}{ 1} \pm \frac{ 110}{ 1} \pm \frac{ 165}{ 1} \pm \frac{ 330}{ 1} ~~ \end{aligned} $$

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( -10 \right) = 0 $ so $ x = -10 $ 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+10 }$

$$ \frac{ x^3+22x^2+153x+330}{ x+10} = x^2+12x+33 $$

Step 2:

The next rational root is $ x = -10 $

$$ \frac{ x^3+22x^2+153x+330}{ x+10} = x^2+12x+33 $$

Step 3:

The solutions of $ x^2+12x+33 = 0 $ are: $ x = -6-\sqrt{ 3 } ~ \text{and} ~ x = -6+\sqrt{ 3 }$.

You can use step-by-step quadratic equation solver to see a detailed explanation on how to solve this quadratic equation.

Polynomial Roots Calculator