Find roots of polynomial $$ p(x) = 8x^5+32x^3+x^2+4 $$

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

$$ \begin{aligned}x_1 &= -\frac{ 1 }{ 2 }\\[1 em]x_2 &= 0.25+0.433i\\[1 em]x_3 &= 0.25-0.433i\\[1 em]x_4 &= 2i\\[1 em]x_5 &= -2i \end{aligned} $$

Step 1:

Use rational root test to find out that the $ \color{blue}{ x = -\dfrac{ 1 }{ 2 } } $ is a root of polynomial $ 8x^5+32x^3+x^2+4 $.

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

The leading coefficient is $ \color{red}{ 8 }$, with factors of 1, 2, 4 and 8.

The POSSIBLE zeroes are:

$$ \begin{aligned} \dfrac{\color{blue}{p}}{\color{red}{q}} = & \dfrac{ \text{ factors of 4 }}{\text{ factors of 8 }} = \pm \dfrac{\text{ ( 1, 2, 4 ) }}{\text{ ( 1, 2, 4, 8 ) }} = \\[1 em] = & \pm \frac{ 1}{ 1} \pm \frac{ 2}{ 1} \pm \frac{ 4}{ 1} ~~ \pm \frac{ 1}{ 2} \pm \frac{ 2}{ 2} \pm \frac{ 4}{ 2} ~~ \pm \frac{ 1}{ 4} \pm \frac{ 2}{ 4} \pm \frac{ 4}{ 4} ~~ \pm \frac{ 1}{ 8} \pm \frac{ 2}{ 8} \pm \frac{ 4}{ 8} ~~ \end{aligned} $$

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( -\dfrac{ 1 }{ 2 } \right) = 0 $ so $ x = -\dfrac{ 1 }{ 2 } $ 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}{ 2x+1 }$

$$ \frac{ 8x^5+32x^3+x^2+4}{ 2x+1} = 4x^4-2x^3+17x^2-8x+4 $$

Step 2:

The next rational root is $ x = -\dfrac{ 1 }{ 2 } $

$$ \frac{ 8x^5+32x^3+x^2+4}{ 2x+1} = 4x^4-2x^3+17x^2-8x+4 $$

Step 3:

Polynomial $ 4x^4-2x^3+17x^2-8x+4 $ has no rational roots that can be found using Rational Root Test, so the roots were found using quartic formulas.

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