Find roots of polynomial $$ p(x) = d^5-5d^4+7d^3+d^2-8d+4 $$

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

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

Step 1:

Use rational root test to find out that the $ \color{blue}{ d = 1 } $ is a root of polynomial $ d^5-5d^4+7d^3+d^2-8d+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 a single factor of 1, 2 and 4.

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

The POSSIBLE zeroes are:

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

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( 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}{ d-1 }$

$$ \frac{ d^5-5d^4+7d^3+d^2-8d+4}{ d-1} = d^4-4d^3+3d^2+4d-4 $$

Step 2:

The next rational root is $ d = 1 $

$$ \frac{ d^5-5d^4+7d^3+d^2-8d+4}{ d-1} = d^4-4d^3+3d^2+4d-4 $$

Step 3:

The next rational root is $ d = 2 $

$$ \frac{ d^4-4d^3+3d^2+4d-4}{ d-2} = d^3-2d^2-d+2 $$

Step 4:

The next rational root is $ d = -1 $

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

Step 5:

The next rational root is $ d = 1 $

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

Step 6:

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

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

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