Intereting Posts

Calculate a multiple sum of inverse integers.
What is the function given by $\sum_{n=0}^\infty \binom{b+2n}{b+n} x^n$, where $b\ge 0$, $|x| <1$
Proving the number of iterations in the Euclidean algorithm
Closed form for $\int_1^\infty\int_0^1\frac{\mathrm dy\,\mathrm dx}{\sqrt{x^2-1}\sqrt{1-y^2}\sqrt{1-y^2+4\,x^2y^2}}$
Norm of the Resolvent
What is “Bourbaki's style in mathematics”?
Horizontal and vertical tangent space of Orthogonal group
Integrate and measure problem.
Example of linear parabolic PDE that blows up
Conditional Expectation a Decreasing Function Implies Covariance is nonpositive
Orthogonal Decomposition
Density of odd numbers in a sequence relating base 2 and base 3 expansion
Mollifiers: Asymptotic Convergence vs. Mean Convergence
Proving ${n \choose p} \equiv \Bigl \ (\text{mod} \ p)$
Number of reflection symmetries of a basketball

My friend asked me what the roots of $y=x^3+x^2-2x-1$ was.

I didn’t really know and when I graphed it, it had no integer solutions. So I asked him what the answer was, and he said that the $3$ roots were $2\cos\left(\frac {2\pi}{7}\right), 2\cos\left(\frac {4\pi}{7}\right)$ and $2\cos\left(\frac {8\pi}{7}\right)$.

Question:How would you get the roots without using a computer such as Mathematica? Can other equations have roots in Trigonometric forms?

- Prove if $n^2$ is even, then $n$ is even.
- prove that $\frac{(2n)!}{(n!)^2}$ is even if $n$ is a positive integer
- What is the standard interpretation of order of operations for the basic arithmetic operations?
- Why does the discriminant of a cubic polynomial being less than $0$ indicate complex roots?
- Finding an $f(x)$ that satisfies $f(f(x)) = 4 - 3x$
- Simplifying factorials: $\frac{(n-1)!}{(n-2)!}$

Anything helps!

- How to compute the integral $\int_{-\infty}^\infty e^{-x^2/2}\,dx$?
- Proving that $\frac{\csc\theta}{\cot\theta}-\frac{\cot\theta}{\csc\theta}=\tan\theta\sin\theta$
- prove inequation
- simple/dumb logarithmic conversion question
- How do calculators handle $\pi$?
- Algorithm/Formula to compute adding and/or removing compound and/or non-compound percentages from a value?
- Why can/do we multiply all terms of a divisor with polynomial long division?
- When the quadratic formula has square root of zero, how to proceed?
- Prove that $\sup(S)=1$ if $S=\{x \in \mathbb{R}| x^2 < x\}$
- If $\alpha = \frac{2\pi}{7}$ then the find the value of $\tan\alpha .\tan2\alpha +\tan2\alpha \tan4\alpha +\tan4\alpha \tan\alpha.$

Let $p(x) = x^3+x^2-2x-1$, we have $$p(t + t^{-1}) = t^3 + t^2 + t + 1 + t^{-1} + t^{-2} + t^{-3} = \frac{t^7-1}{t^3(t-1)}$$

The RHS has roots of the form $t = e^{\pm \frac{2k\pi}{7}i}$

( coming from the $t^7 – 1$ factor in numerator )

for $k = 1,2,3$. So $p(x)$ has roots of the form $$e^{\frac{2k\pi}{7} i} + e^{-\frac{2k\pi}{7} i} = 2\cos\left(\frac{2 k\pi}{7}\right)$$ for $k = 1,2,3$.

Consider the equation $$\cos4\theta=\cos3\theta$$ whose roots are $$\theta=n\cdot\frac{2\pi}{7}$$

Representing this as a polynomial in $c=\cos\theta$, we have $$8c^4-4c^3-8c^2+3c+1=0$$

$$\Rightarrow (c-1)(8c^3+4c^2-4c-1)=0$$

Now write $x=2c$ and we see that the polynomial equation $$x^3+x^2-2x-1=0$$ has roots as stated in your question. Note that $$2\cos\frac{6\pi}{7}=2\cos\frac{8\pi}{7}$$

Set $x=t+t^{-1}$. Then the equation becomes

$$

t^3+3t+3t^{-1}+t^{-3}+t^2+2+t^{-2}-2t-2t^{-1}-1=0

$$

and, multiplying by $t^3$,

$$

t^6+t^5+t^4+t^3+t^2+t+1=0

$$

and it should be now clear what the solutions are. For each root there’s another one giving the same solution in $x$.

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- Why does this $u$-substitution zero out my integral?
- Estimating the number of integers relatively prime to $6$ between $1$ and some integer $x$?
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- Find the Jacobian
- Polynomials irreducible over $\mathbb{Q}$ but reducible over $\mathbb{F}_p$ for every prime $p$
- Proving there don't exist $F(x), G(x)$ such that $1^{-1}+2^{-1}+3^{-1}+\cdots+n^{-1}={F(n)}/{G(n)}$
- Complement of closed dense set
- Is the function $F(x,y)=1−e^{−xy}$ $0 ≤ x$, $y < ∞$, the joint cumulative distribution function of some pair of random variables?