Find roots of polynomial $$ p(x) = -4x^4+36x^3-117x^2+162x-81 $$

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

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

Step 1:

Use rational root test to find out that the $ \color{blue}{ x = 3 } $ is a root of polynomial $ -4x^4+36x^3-117x^2+162x-81 $.

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}{ 81 } $, with factors of 1, 3, 9, 27 and 81.

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

The POSSIBLE zeroes are:

$$ \begin{aligned} \dfrac{\color{blue}{p}}{\color{red}{q}} = & \dfrac{ \text{ factors of 81 }}{\text{ factors of 4 }} = \pm \dfrac{\text{ ( 1, 3, 9, 27, 81 ) }}{\text{ ( 1, 2, 4 ) }} = \\[1 em] = & \pm \frac{ 1}{ 1} \pm \frac{ 3}{ 1} \pm \frac{ 9}{ 1} \pm \frac{ 27}{ 1} \pm \frac{ 81}{ 1} ~~ \pm \frac{ 1}{ 2} \pm \frac{ 3}{ 2} \pm \frac{ 9}{ 2} \pm \frac{ 27}{ 2} \pm \frac{ 81}{ 2} ~~ \pm \frac{ 1}{ 4} \pm \frac{ 3}{ 4} \pm \frac{ 9}{ 4} \pm \frac{ 27}{ 4} \pm \frac{ 81}{ 4} ~~ \end{aligned} $$

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}{ x-3 }$

$$ \frac{ -4x^4+36x^3-117x^2+162x-81}{ x-3} = -4x^3+24x^2-45x+27 $$

Step 2:

The next rational root is $ x = 3 $

$$ \frac{ -4x^4+36x^3-117x^2+162x-81}{ x-3} = -4x^3+24x^2-45x+27 $$

Step 3:

The next rational root is $ x = \dfrac{ 3 }{ 2 } $

$$ \frac{ -4x^3+24x^2-45x+27}{ 2x-3} = -2x^2+9x-9 $$

Step 4:

The next rational root is $ x = 3 $

$$ \frac{ -2x^2+9x-9}{ x-3} = -2x+3 $$

Step 5:

To find the last zero, solve equation $ -2x+3 = 0 $

$$ \begin{aligned} -2x+3 & = 0 \\[1 em] -2 \cdot x & = -3 \\[1 em] x & = - \frac{ 3 }{ -2 } \\[1 em] x & = \frac{ 3 }{ 2 } \end{aligned} $$

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