About the number of real roots

Question

I want to solve this equation:

$$t^{10}-t^{9}+ t^{8}- t^{7}+ t^{6}- t^{5}+ t^{4}- t^{3}+ t^{2}- t+1=0$$

with respect to $t$. But I have not a good idea to start. Hence, I am asking about the number of real roots.

Can we deduce the same result for a polynomial of the form:

$$at^{10}-bt^{9}+ ct^{8}-d t^{7}+e t^{6}- ft^{5}+ gt^{4}- ht^{3}+ lt^{2}-m t+r=0$$

where all the coefficient are real and positive.

Answer 1

Let $p(t)$ denote your polynomial. Then it is not hard to see that $$(1+t)p(t)=t^{11}+1,$$ which clearly has $-1$ as its only real root. But $p(-1)=11$, so $p(t)$ has no real roots.

This also shows that over the complex numbers, the roots of $p(t)$ all satisfy $t^{11}=-1$. By Euler's formula we have $$e^{\pi i}=-1,$$ so the roots are all of the form $\exp\left(\tfrac{k}{11}\pi i\right)$ for some integer $k$.

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Note that $p(t)$ is in fact the 22nd cyclotomic polynomial; the roots of the $n$-th cyclotomic are the primitive $n$-th roots of unity, which are not real for $n>2$.

Answer 2

Hint:

Use the high-school identity $$t^{2n+1}+1=(t+1)(t^{2n}-t^{2n-1}+t^{2n-2}-\dots+t^2-t+1).$$

What can you conclude for the roots of your polynomial?

Answer 3

Let $t\geq1$.

Thus, $$t^{10}-t^9+t^8-t^7+t^6-t^5+t^4-t^3+t^2-t+1=$$ $$=t^9(t-1)+t^7(t-1)+t^5(t-1)+t^3(t-1)+t(t-1)+1>0.$$ Let $0<t<1.$

Thus, $$$t^{10}-t^9+t^8-t^7+t^6-t^5+t^4-t^3+t^2-t+1=$$ $$1-t+t^2(1-t)+t^4(1-t)+t^6(1-t)+t^8(1-t)+t^{10}>0.$$ Let $t\leq0.$

Thus, it's obvious that $$t^{10}-t^9+t^8-t^7+t^6-t^5+t^4-t^3+t^2-t+1>0.$$ Id est, our equation has no real roots.

Answer 4

Q: Can we deduce the same result for a polynomial of the form:

$$at^{10}-bt^{9}+ ct^{8}-d t^{7}+e t^{6}- ft^{5}+ gt^{4}- ht^{3}+ lt^{2}-m t+r=0$$

where all coefficients are real and positive$\,\mathbf ?$

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Not at all! Let

$$ C_1\ \ldots\ C_{10}\ > 0 $$

be ten real positive constants. Let

$$ f(t)\ :\ \prod_{k=1}^{10}(t-C_k) $$

This polynomial $f$ will have all respective ten coefficient $\ a\ldots\ r\ $ positive, as required, but this time our polynomial has ten real roots, namely $\ C_1\ \ldots\ C_{10}\ $ (some of them can be multiple roots when there are some repetitions among constants $C_k$).