Tagged Questions

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How to obtain the number of real valued zeroes of a polynomial?

While I know there's no analytical formula for the roots of a general polynomial of degree five and higher, I wonder whether there is at least something like a parabola's discriminant to determine how ...
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Roots of this trigonometric polynomial

Let $f:[0,2\pi) \rightarrow \mathbb{R}$ with $f(x):=\sum_{n=0}^{k}a_n \left(1+\cos(x)\right)^n$ for arbitrary $a_n$ with $a_k \neq 0$. My question is: What is the maximum number of zeros that this ...
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Using intermediate value arguments at limits rather than finding explicit bounds

Again, I apologize for what looks like a very narrow question. But there's possibly a general principal at work here that I'm not grasping. I understand the answer provided for exercise 3 in chapter ...
Given a cubic and quadratic share a root, prove $(ac-b^{2})(bd-c^{2})\geq 0$
Here is an interesting problem. Perhaps someone would be so kind as to give me a shove in the right direction?. If $ax^{3}+3bx^{2}+3cx+d$ and $ax^{2}+2bx+c$ share a common root, then prove that ...