Find roots of polynomial $$ p(x) = 3x^3+x^2-62x+40 $$

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

$$ \begin{aligned}x_1 &= 4\\[1 em]x_2 &= -5\\[1 em]x_3 &= \frac{ 2 }{ 3 } \end{aligned} $$

Step 1:

Use rational root test to find out that the $ \color{blue}{ x = 4 } $ is a root of polynomial $ 3x^3+x^2-62x+40 $.

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}{ 40 } $, with factors of 1, 2, 4, 5, 8, 10, 20 and 40.

The leading coefficient is $ \color{red}{ 3 }$, with factors of 1 and 3.

The POSSIBLE zeroes are:

$$ \begin{aligned} \dfrac{\color{blue}{p}}{\color{red}{q}} = & \dfrac{ \text{ factors of 40 }}{\text{ factors of 3 }} = \pm \dfrac{\text{ ( 1, 2, 4, 5, 8, 10, 20, 40 ) }}{\text{ ( 1, 3 ) }} = \\[1 em] = & \pm \frac{ 1}{ 1} \pm \frac{ 2}{ 1} \pm \frac{ 4}{ 1} \pm \frac{ 5}{ 1} \pm \frac{ 8}{ 1} \pm \frac{ 10}{ 1} \pm \frac{ 20}{ 1} \pm \frac{ 40}{ 1} ~~ \pm \frac{ 1}{ 3} \pm \frac{ 2}{ 3} \pm \frac{ 4}{ 3} \pm \frac{ 5}{ 3} \pm \frac{ 8}{ 3} \pm \frac{ 10}{ 3} \pm \frac{ 20}{ 3} \pm \frac{ 40}{ 3} ~~ \end{aligned} $$

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 }$

$$ \frac{ 3x^3+x^2-62x+40}{ x-4} = 3x^2+13x-10 $$

Step 2:

The next rational root is $ x = 4 $

$$ \frac{ 3x^3+x^2-62x+40}{ x-4} = 3x^2+13x-10 $$

Step 3:

The next rational root is $ x = -5 $

$$ \frac{ 3x^2+13x-10}{ x+5} = 3x-2 $$

Step 4:

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

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

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