Find roots of polynomial $$ p(x) = 4x^3+21x^2-126x+81 $$

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

$$ \begin{aligned}x_1 &= 3\\[1 em]x_2 &= -9\\[1 em]x_3 &= \frac{ 3 }{ 4 } \end{aligned} $$

Step 1:

Use rational root test to find out that the $ \color{blue}{ x = 3 } $ is a root of polynomial $ 4x^3+21x^2-126x+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 }$

$$ \frac{ 4x^3+21x^2-126x+81}{ x-3} = 4x^2+33x-27 $$

Step 2:

The next rational root is $ x = 3 $

$$ \frac{ 4x^3+21x^2-126x+81}{ x-3} = 4x^2+33x-27 $$

Step 3:

The next rational root is $ x = -9 $

$$ \frac{ 4x^2+33x-27}{ x+9} = 4x-3 $$

Step 4:

To find the last zero, solve equation $ 4x-3 = 0 $

$$ \begin{aligned} 4x-3 & = 0 \\[1 em] 4 \cdot x & = 3 \\[1 em] x & = \frac{ 3 }{ 4 } \end{aligned} $$

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