# Polynomials sprint: How to solve a seemingly disgusting polynomial

On page 248, it is shown how to solve the polynomial $$x^4+x^3+x^2+x+1=0$$. In this note, I will explain how to solve the polynomial $x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1=0$ First we multiply both sides by $$(x-1)$$ to get $x^9-1=0\implies x^9=1$ Then since $$x^9=1$$, we can divide some terms by $$x^9$$ $x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1=$ $\dfrac{x^8}{x^9}+\dfrac{x^7}{x^9}+\dfrac{x^6}{x^9}+\dfrac{x^5}{x^9}+x^4+x^3+x^2+x+1=$ $\dfrac{1}{x}+\dfrac{1}{x^2}+\dfrac{1}{x^3}+\dfrac{1}{x^4}+x^4+x^3+x^2+x+1$ $\left(x+\dfrac{1}{x}\right)+\left(x^2+\dfrac{1}{x^2}\right)+\left(x^3+\dfrac{1}{x^3}\right)+\left(x^4+\dfrac{1}{x^4}\right)+1=0$ Now we let $$y=x+\dfrac{1}{x}$$ so we have $(y)+(y^2-2)+(y^3-3y)+((y^2-2)^2-2)+1=0$ which after simplification becomes $y^4+y^3-3y^2-2y+1=0$ Using the Rational Root Theorem (discussed on page 246) we quickly find that $$y=-1$$ is a root and factor the polynomial as $(y+1)(y^3-3y+1)$ Now the rest is simple! We can use the cubic solving method discussed on pages 247-248 to find the roots of $$y^3-3y+1$$ Then simple quadratic bashing gives us our roots!

Note by Nathan Ramesh
4 years, 2 months ago

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Or, we can use the 9th roots of unity.

- 4 years, 2 months ago

Yes, what happened to the classic roots of unity? Roots are just $$e^{2ki\pi/9}$$with $$k=1\to 8$$.

- 4 years, 2 months ago

Except all roots of unity give you is $$e^{\frac{2\pi i}{9}}$$ and not a numerical answer with radicals. The method here allows you to compute $$\sin {40^{\circ}}$$

- 4 years, 2 months ago

Or you can also just do some Euler's formula with $$e^{\dfrac{2\pi i}{9}}$$.

- 4 years, 2 months ago

ummmm which euler's formula?

- 4 years, 2 months ago

$$e^{i\phi}$$ = cos$$\phi$$ + $${i}$$sin$$\phi$$ ... substituting $$\pi$$ gives $$e^{i\pi}$$ = -1, where $${i}$$ = $$\sqrt{-1}$$

- 3 years, 8 months ago

Where i can find this book

- 3 years, 8 months ago

$$e^{i\theta}=\text{cis}(\theta)$$

- 4 years, 2 months ago

How does that let you solve for $$\sin{40^{\circ}}$$

- 4 years, 2 months ago

wow

- 4 years, 2 months ago

:o

- 4 years, 2 months ago

2 Daniels

- 4 years, 2 months ago

Oh God, this is not good. One Daniel is enough for me, but two? Double Trouble.

- 4 years, 2 months ago

2 Lius

- 4 years, 2 months ago

2 14-year-old-Daniel-Lius. Enough to get a conversation off-topic.

- 4 years, 2 months ago

2 14-year-old-Daniel-Lius-residing-in-USA. conversation closed.

- 4 years, 2 months ago

Where are all other pages?

- 3 years, 8 months ago

Yes I used the same to expand cosnx

- 3 years, 8 months ago

@Nathan Ramesh Can you add this to the Roots of Unity Applications Wiki page?

Select "Write a summary", and then copy-paste your text into it (with minor formatting adjustments if relevant. Thanks!

Staff - 3 years, 11 months ago

Done! I put it at the bottom. Let me know if it is bugged (I posted it from my phone). Thanks!

- 3 years, 11 months ago