# Tagged Questions

Questions about the set of values at which a given function evaluates to zero. For questions about "square roots", "cube roots", and such, consider using the (radicals) and (arithmetic) tag. For questions about roots of Lie algebras, use the (lie-algebra) tag instead.

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### Prove that the unique zeros of $f(x,y)=a x +(1-a)y+xy$ when $x,y\in[0,1]$, is $x=y=0$.

Prove that the unique zeros of the two-variables function: $$f(x,y)=a x +(1-a)y+xy$$ when $x,y\in[0,1]$, is $x=y=0$. Here, $a$ is a parameter between 0 and 1. I have no idea where to start. Any ...
533 views

### Relation between real roots of a polynomial and real roots of its derivative

I have this question which popped in my mind while solving questions of maxima and minima. First Case:Let $f(x)$ be an $n$ degree polynomial which has $r$ real roots. Using this can we say anything ...
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### Roots of $f(x)=x-2+\frac{a-3}{x}$

I wanted to find the values of (a) for which the function $f(x)=x-2+\frac{a-3}{x}$ has more than one root. I know that the equation needs to be set equal to zero, from that step onward I have no idea ...
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### Why isn't the square root is cancelled in this formula?

$\sqrt{\sum\limits_{i=1}^M \vec{V^2_d}(d)}$ This is the formula of the Euclidean length of a vector in the vector space. The vector $V$ has a power of 2 so it is $V^2$. Why isn't the square root of ...
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### Is it possible to find solutions to polynomials purely by calculus and without iteration?

I know this may sound peculiar, but I was wondering if any mathematicians have found a method to finding roots purely through calculus without iteration. I can't imagine that such a method exists for ...
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### Real Roots of Complex Quadratic Equation - (Kasana's first example)

I recently bought H.S. Kasana's Complex Variables. It seems quite interesting, and a little harder for me than I had expected, though I should be able to get through it if I take my time. ...
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### Equilibrium Points for 8th Degree Polynomial

I have an 8th degree polynomial that I need the zeros for. Is there even a way to explicitly solve one? Its for a diff equations review. I need to sketch the phase line, which is a breeze once I get ...
536 views

### Solving following quartic equation

Solve in $\mathbb{R}$ : $$(x^2+2)^2+8x^2=6x (x^2+2)$$ My try: I tried to make the graph by calculating values for $x=1, 2, 3, 4$ and I found that the function is positive at $x=0$ but negative at ...
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### (Discriminant) For which values of k will the equation g(x) = x + k have two real roots that are of opposite signs?

I am currently in Grade 12 and came across the following question in a past paper: $$g(x) = \frac{2}{x+1}+1$$ The question asks: For which values of k will the equation $g(x) = x + k$ have two real ...
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### Finding Zeros of cubic without using fzero or roots in Matlab [closed]

okay, so I've modifies my code a bit. ...
331 views

### Number of real roots

Find number of real roots of the equation $$3^{|x|}-|2-|x||=1$$ My try:I have tried to remove the modulas by assuming x in some intervals and moved the linear part to RHS and drawn the rough graph ...
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### Locating the roots of a cubic polynomial.

Given a cubic polynomial $f(x) = ax^{3} + bx^{2} + cx +d$ with arbitrary real coefficients and $a\neq 0$. Is there an easy test to determine when all the real roots of $f$ are negative? The Routh-...
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### What is the relationship between the concept of a square root and a number's prime factorization?

Essentially what I am asking is if there is some kind of correlation between a number such as √385 and it's factorization (which is 5,7,11). Is it possible to use a number's (especially very large ...
215 views

### Matlab Coding finding zeros without using fzero or roots function

So i am a completely new at Matlab. I'm basically suppose to develop a function in Matlab that finds the zeros of a cubic polynomial. real and complex. I'm pasting below what I have so far. I started ...
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### Find all the values of $k$, if any, such that $f=t^4+2t^3-3t^2+2kt+k^2$ is divisible by $g=t+2$ in $\mathbb{Z}_{7}[t]$

Find all the values of $k$, if any, such that $f=t^4+2t^3-3t^2+2kt+k^2$ is divisible by $g=t+2$ in $\mathbb{Z}_{7}[t]$. I solve it in the normal way but I do not sure that my way is correct or not ...
337 views

### How to find the roots of a 2 variable polynomial of 2nd degree?

The following polynomial is just an example: $$(3-3y)(x^2-y)$$ and is what does it mean to find the critical points of this polynomial? These are the maxima minima. Are they always concerned with ...
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### Find Zeros / Factors of a polynomial

I have been told that to find factors of a polynomial (nth degree) we have to find the factors of constant term and that of coefficient of leading term of the polynomial in concern. The possible ...
121 views

### Solve $x+y+z=1; x^2+y^2+z^2=35; x^3+y^3+z^3=97$

It may be surprising that I can't get any analytical way of verifying that one of the solutions of $$x+y+z=1$$ $$x^2+y^2+z^2=35$$ $$x^3+y^3+z^3=97$$ is $x=-1, y=-3$ and $z=5$. Although it may be ...
116 views

### How to count the real roots of a quartic equation?

Suppose I have a quartic equation with real coefficients, such as: $$a x^4 +b x^3+c x^2+d x +e=0$$ I want to know the number of its real roots. Search engines lead me to symbolic expressions for all ...
123 views

### Solving for $x$ : $a^x+b^x=c$

Well the question is to solve for $x$ in $$a^x+b^x=c \tag{a,b,c are constants}$$ Well as of me, I tried to put $\ln{}$ on both sides which does not seem to help. Apart from this I don't seem to have ...
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### Find coefficients of polynomials $f(x)=x^2+ax+b$ and $g(x)=x^2+cx+d$ $(a,b,c,d \in \mathbb{R})$

Roots of polynomial $f(x)=x^2+ax+b$ are cubes of the roots of polynomial $g(x)=x^2+cx+d$. Sum and product of roots of polynomial $g(x)$ are equal. Find coefficients $a,b,c,d$ so that polynomial $f(x)$ ...
128 views

### Negative roots of a cubic equation

Under what conditions will the cubic equation $ax^3 + bx^2 + cx + d$ where $a,b,c,d \in \mathbb R$ yield roots which have negative real parts? (All roots must have negative real parts) Motivation: I ...
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### Showing that the roots of the quadratic are real

If $x^2+bx+c=0$ has real roots, show that the roots of the equation $x^2+bx+c(x+a)(2x+b)=0$ are real for all real values of $a$. I could do it by standard way by proving determinant is postive. But,...
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### Finding the root for a multivariate function.

Background In a practical problem I need to find the solution to: $$f(\bar{x}) - \bar{p} = \bar{0}$$ where $f : \mathbb{R}^2 \rightarrow \mathbb{R}^2$. I don't know the exact expression for $f$ ...
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### How to do polynomial composition/substitution? (Vincent-Alesina-Galuzzi)

For the polynomial $$p(x) = \sum_{i=0}^n c_i x^i,$$ of real coefficients and real variable, obtain the coefficient of $$q(x) = \left(1 + x\right)^n p\left( \frac{a + b x}{1 + x} \right),$$ as ...
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### Is there any notation for general $n$-th root $r$ such that $r^n=x$?
As we know that the notation for the $n$-th principal root is $\sqrt[n]{x}$ or $x^{1/n}$. But the principal root is not always the only possible root, e.g. for even $n$ and positive $x$ the principal ...
### Prove that $f(x)=m$ has three distinct real roots for $m\in(0,8)$
We have $f:\mathbb{R}\rightarrow\mathbb{R},f(x)=x^5-5x+4$ and we need to show that $\forall m\in(0,8)$, $f(x)=m$ has three distinct real roots. How can I prove it?