Find roots of polynomial $$ p(x) = -x^3+26x^2-205x+468 $$

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

$$ \begin{aligned}x_1 &= 4\\[1 em]x_2 &= 9\\[1 em]x_3 &= 13 \end{aligned} $$

Step 1:

Use rational root test to find out that the $ \color{blue}{ x = 4 } $ is a root of polynomial $ -x^3+26x^2-205x+468 $.

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}{ 468 } $, with a single factor of 1, 2, 3, 4, 6, 9, 12, 13, 18, 26, 36, 39, 52, 78, 117, 156, 234 and 468.

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

The POSSIBLE zeroes are:

$$ \begin{aligned} \dfrac{\color{blue}{p}}{\color{red}{q}} = & \dfrac{ \text{ factors of 468 }}{\text{ factors of 1 }} = \pm \dfrac{\text{ ( 1, 2, 3, 4, 6, 9, 12, 13, 18, 26, 36, 39, 52, 78, 117, 156, 234, 468 ) }}{\text{ ( 1 ) }} = \\[1 em] = & \pm \frac{ 1}{ 1} \pm \frac{ 2}{ 1} \pm \frac{ 3}{ 1} \pm \frac{ 4}{ 1} \pm \frac{ 6}{ 1} \pm \frac{ 9}{ 1} \pm \frac{ 12}{ 1} \pm \frac{ 13}{ 1} \pm \frac{ 18}{ 1} \pm \frac{ 26}{ 1} \pm \frac{ 36}{ 1} \pm \frac{ 39}{ 1} \pm \frac{ 52}{ 1} \pm \frac{ 78}{ 1} \pm \frac{ 117}{ 1} \pm \frac{ 156}{ 1} \pm \frac{ 234}{ 1} \pm \frac{ 468}{ 1} ~~ \end{aligned} $$

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( 4 \right) = 0 $ so $ x = 4 $ 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-4 }$

$$ \frac{ -x^3+26x^2-205x+468}{ x-4} = -x^2+22x-117 $$

Step 2:

The next rational root is $ x = 4 $

$$ \frac{ -x^3+26x^2-205x+468}{ x-4} = -x^2+22x-117 $$

Step 3:

The next rational root is $ x = 9 $

$$ \frac{ -x^2+22x-117}{ x-9} = -x+13 $$

Step 4:

To find the last zero, solve equation $ -x+13 = 0 $

$$ \begin{aligned} -x+13 & = 0 \\[1 em] -1 \cdot x & = -13 \\[1 em] x & = - \frac{ 13 }{ -1 } \\[1 em] x & = 13 \end{aligned} $$

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