Find roots of polynomial $$ p(x) = 7x^4-8x^3-42x^2+7x+6 $$

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

$$ \begin{aligned}x_1 &= 3\\[1 em]x_2 &= -2\\[1 em]x_3 &= 0.4561\\[1 em]x_4 &= -0.3132 \end{aligned} $$

Step 1:

Use rational root test to find out that the $ \color{blue}{ x = 3 } $ is a root of polynomial $ 7x^4-8x^3-42x^2+7x+6 $.

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}{ 6 } $, with factors of 1, 2, 3 and 6.

The leading coefficient is $ \color{red}{ 7 }$, with factors of 1 and 7.

The POSSIBLE zeroes are:

$$ \begin{aligned} \dfrac{\color{blue}{p}}{\color{red}{q}} = & \dfrac{ \text{ factors of 6 }}{\text{ factors of 7 }} = \pm \dfrac{\text{ ( 1, 2, 3, 6 ) }}{\text{ ( 1, 7 ) }} = \\[1 em] = & \pm \frac{ 1}{ 1} \pm \frac{ 2}{ 1} \pm \frac{ 3}{ 1} \pm \frac{ 6}{ 1} ~~ \pm \frac{ 1}{ 7} \pm \frac{ 2}{ 7} \pm \frac{ 3}{ 7} \pm \frac{ 6}{ 7} ~~ \end{aligned} $$

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 }$

$$ \frac{ 7x^4-8x^3-42x^2+7x+6}{ x-3} = 7x^3+13x^2-3x-2 $$

Step 2:

The next rational root is $ x = 3 $

$$ \frac{ 7x^4-8x^3-42x^2+7x+6}{ x-3} = 7x^3+13x^2-3x-2 $$

Step 3:

The next rational root is $ x = -2 $

$$ \frac{ 7x^3+13x^2-3x-2}{ x+2} = 7x^2-x-1 $$

Step 4:

The solutions of $ 7x^2-x-1 = 0 $ are: $ x = \dfrac{ 1 }{ 14 }-\dfrac{\sqrt{ 29 }}{ 14 } ~ \text{and} ~ x = \dfrac{ 1 }{ 14 }+\dfrac{\sqrt{ 29 }}{ 14 }$.

You can use step-by-step quadratic equation solver to see a detailed explanation on how to solve this quadratic equation.

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