Find roots of polynomial $$ p(x) = 4d^5-23d^3-33d^2-17d-3 $$

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

$$ \begin{aligned}d_1 &= 3\\[1 em]d_2 &= -1\\[1 em]d_3 &= -\frac{ 1 }{ 2 }\\[1 em]d_4 &= -1\\[1 em]d_5 &= -\frac{ 1 }{ 2 } \end{aligned} $$

Step 1:

Use rational root test to find out that the $ \color{blue}{ d = 3 } $ is a root of polynomial $ 4d^5-23d^3-33d^2-17d-3 $.

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}{ 3 } $, with factors of 1 and 3.

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

The POSSIBLE zeroes are:

$$ \begin{aligned} \dfrac{\color{blue}{p}}{\color{red}{q}} = & \dfrac{ \text{ factors of 3 }}{\text{ factors of 4 }} = \pm \dfrac{\text{ ( 1, 3 ) }}{\text{ ( 1, 2, 4 ) }} = \\[1 em] = & \pm \frac{ 1}{ 1} \pm \frac{ 3}{ 1} ~~ \pm \frac{ 1}{ 2} \pm \frac{ 3}{ 2} ~~ \pm \frac{ 1}{ 4} \pm \frac{ 3}{ 4} ~~ \end{aligned} $$

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( 3 \right) = 0 $ so $ x = 3 $ 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}{ d-3 }$

$$ \frac{ 4d^5-23d^3-33d^2-17d-3}{ d-3} = 4d^4+12d^3+13d^2+6d+1 $$

Step 2:

The next rational root is $ d = 3 $

$$ \frac{ 4d^5-23d^3-33d^2-17d-3}{ d-3} = 4d^4+12d^3+13d^2+6d+1 $$

Step 3:

The next rational root is $ d = -1 $

$$ \frac{ 4d^4+12d^3+13d^2+6d+1}{ d+1} = 4d^3+8d^2+5d+1 $$

Step 4:

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

$$ \frac{ 4d^3+8d^2+5d+1}{ 2d+1} = 2d^2+3d+1 $$

Step 5:

The next rational root is $ d = -1 $

$$ \frac{ 2d^2+3d+1}{ d+1} = 2d+1 $$

Step 6:

To find the last zero, solve equation $ 2d+1 = 0 $

$$ \begin{aligned} 2d+1 & = 0 \\[1 em] 2 \cdot d & = -1 \\[1 em] d & = - \frac{ 1 }{ 2 } \end{aligned} $$

Polynomial Roots Calculator