Find roots of polynomial $$ p(x) = x^6-7x^4+14x^2-8 $$

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

$$ \begin{aligned}x_1 &= 1\\[1 em]x_2 &= 2\\[1 em]x_3 &= -1\\[1 em]x_4 &= -2\\[1 em]x_5 &= \sqrt{ 2 }\\[1 em]x_6 &= -\sqrt{ 2 } \end{aligned} $$

Step 1:

Use rational root test to find out that the $ \color{blue}{ x = 1 } $ is a root of polynomial $ x^6-7x^4+14x^2-8 $.

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}{ 8 } $, with a single factor of 1, 2, 4 and 8.

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

The POSSIBLE zeroes are:

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

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( 1 \right) = 0 $ so $ x = 1 $ 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-1 }$

$$ \frac{ x^6-7x^4+14x^2-8}{ x-1} = x^5+x^4-6x^3-6x^2+8x+8 $$

Step 2:

The next rational root is $ x = 1 $

$$ \frac{ x^6-7x^4+14x^2-8}{ x-1} = x^5+x^4-6x^3-6x^2+8x+8 $$

Step 3:

The next rational root is $ x = 2 $

$$ \frac{ x^5+x^4-6x^3-6x^2+8x+8}{ x-2} = x^4+3x^3-6x-4 $$

Step 4:

The next rational root is $ x = -1 $

$$ \frac{ x^4+3x^3-6x-4}{ x+1} = x^3+2x^2-2x-4 $$

Step 5:

The next rational root is $ x = -2 $

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

Step 6:

The solutions of $ x^2-2 = 0 $ are: $ x = -\sqrt{ 2 } ~ \text{and} ~ x = \sqrt{ 2 }$.

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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