Find roots of polynomial $$ p(x) = x^{10}-11x^8+41x^6-65x^4+43x^2-9 $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= 1\\[1 em]x_2 &= -1\\[1 em]x_3 &= 0.618\\[1 em]x_4 &= -0.618\\[1 em]x_5 &= -1.3028\\[1 em]x_6 &= 1.3028\\[1 em]x_7 &= 1.618\\[1 em]x_8 &= -1.618\\[1 em]x_9 &= 2.3028\\[1 em]x_{10} &= -2.3028 \end{aligned} $$Explanation
Step 1:
Use rational root test to find out that the $ \color{blue}{ x = 1 } $ is a root of polynomial $ x^{10}-11x^8+41x^6-65x^4+43x^2-9 $.
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}{ 9 } $, with a single factor of 1, 3 and 9.
The leading coefficient is $ \color{red}{ 1 }$, with a single factor of 1.
The POSSIBLE zeroes are:
$$ \begin{aligned} \dfrac{\color{blue}{p}}{\color{red}{q}} = & \dfrac{ \text{ factors of 9 }}{\text{ factors of 1 }} = \pm \dfrac{\text{ ( 1, 3, 9 ) }}{\text{ ( 1 ) }} = \\[1 em] = & \pm \frac{ 1}{ 1} \pm \frac{ 3}{ 1} \pm \frac{ 9}{ 1} ~~ \end{aligned} $$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 }$
$$ \frac{ x^{10}-11x^8+41x^6-65x^4+43x^2-9}{ x-1} = x^9+x^8-10x^7-10x^6+31x^5+31x^4-34x^3-34x^2+9x+9 $$Step 2:
The next rational root is $ x = 1 $
$$ \frac{ x^{10}-11x^8+41x^6-65x^4+43x^2-9}{ x-1} = x^9+x^8-10x^7-10x^6+31x^5+31x^4-34x^3-34x^2+9x+9 $$Step 3:
The next rational root is $ x = -1 $
$$ \frac{ x^9+x^8-10x^7-10x^6+31x^5+31x^4-34x^3-34x^2+9x+9}{ x+1} = x^8-10x^6+31x^4-34x^2+9 $$Step 4:
Polynomial $ x^8-10x^6+31x^4-34x^2+9 $ has no rational roots that can be found using Rational Root Test, so the roots were found using Newton method.