Find roots of polynomial $$ p(x) = 5x^4+31x^3+67x^2-11x-52 $$

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

$$ \begin{aligned}x_1 &= -1\\[1 em]x_2 &= \frac{ 4 }{ 5 }\\[1 em]x_3 &= -3+2i\\[1 em]x_4 &= -3-2i \end{aligned} $$

Step 1:

Use rational root test to find out that the $ \color{blue}{ x = -1 } $ is a root of polynomial $ 5x^4+31x^3+67x^2-11x-52 $.

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}{ 52 } $, with factors of 1, 2, 4, 13, 26 and 52.

The leading coefficient is $ \color{red}{ 5 }$, with factors of 1 and 5.

The POSSIBLE zeroes are:

$$ \begin{aligned} \dfrac{\color{blue}{p}}{\color{red}{q}} = & \dfrac{ \text{ factors of 52 }}{\text{ factors of 5 }} = \pm \dfrac{\text{ ( 1, 2, 4, 13, 26, 52 ) }}{\text{ ( 1, 5 ) }} = \\[1 em] = & \pm \frac{ 1}{ 1} \pm \frac{ 2}{ 1} \pm \frac{ 4}{ 1} \pm \frac{ 13}{ 1} \pm \frac{ 26}{ 1} \pm \frac{ 52}{ 1} ~~ \pm \frac{ 1}{ 5} \pm \frac{ 2}{ 5} \pm \frac{ 4}{ 5} \pm \frac{ 13}{ 5} \pm \frac{ 26}{ 5} \pm \frac{ 52}{ 5} ~~ \end{aligned} $$

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 }$

$$ \frac{ 5x^4+31x^3+67x^2-11x-52}{ x+1} = 5x^3+26x^2+41x-52 $$

Step 2:

The next rational root is $ x = -1 $

$$ \frac{ 5x^4+31x^3+67x^2-11x-52}{ x+1} = 5x^3+26x^2+41x-52 $$

Step 3:

The next rational root is $ x = \dfrac{ 4 }{ 5 } $

$$ \frac{ 5x^3+26x^2+41x-52}{ 5x-4} = x^2+6x+13 $$

Step 4:

The solutions of $ x^2+6x+13 = 0 $ are: $ x = -3+2i ~ \text{and} ~ x = -3-2i$.

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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