# 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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### Find the algebric form of the zeros(roots) of the following polynomial: $\left(\:z^2+iz+2\right)\left(z^3-8i\right)$

Good morning to everyone. I don't know how to find the zeros(roots) of the following polynomial function: $$\left(\:z^2+iz+2\right)\left(z^3-8i\right)$$. What I've tried: The zero(root) of the second ...
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### Why can a quartic polynomial never have three real and one complex root?

It seems that a quartic polynomial (degree $4$) either can have $0$ real, $1$ real, $2$ real, or $4$ real roots, and the rest is complex roots. Why can't it have $3$ real roots and $1$ complex?
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### Existence of solution of equation involving normal distribution

I've tried to show that the following equation has a solution: \begin{equation*} g(x)=\left[1-\left(2\int _{\mu}^{x}f(y)dy\right)^2\right]-8xf(x)\int _{\mu}^{x}f(y)dy=0, \end{equation*} where $f(x)$ ...
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### Are there more quadratics with real roots or more with complex roots? Or the same?

Consider all quadratic equations with real coefficients: $$y=ax^2+bx+c \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,, a,b,c \in \mathbb{R}, \, a≠0$$ I was wondering whether if more of them have real roots, more ...
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### non-complex cubic roots formula?

Suppose we have a cubic equation $$ax^3 +bx^2 +cx +d =0$$ for which we know that all three distinct roots are real. Do we have a formula for them that does not involve complex roots of unity? The ...
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### Complex Conjugate roots with non real coefficients

I understand that a polynomial with real coefficients must have complex conjugate roots (if complex roots exist) Is it possible for a polynomial with non-real coefficients to have complex conjugate ...
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### How to compute equation with exponents?

I want to find $a$, where that term satisfies this equation: $$a + a(1-a) + a(1-a)^2 + \cdots + a(1-a)^{15} = 0.5$$ I could write that as a sum from 0 to 15, but still it is unclear to me how should ...
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### Trigonometric Roots of a Polynomial

After wondering on this question, I wondered how would you be able to find the roots of a polynomial, in the form $y=x^3+ax^2+bx+c$ if they are the sums of cosines? I'm wondering if it can, too, be ...
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### Specific fucnction has 11 different zeros

Let $f : \mathbb{C} \to \mathbb{C}$ be given by $$f(z) = z^{11} + 4 e^{z + 1} - 2$$ Show that $f$ has 11 different zeros in the annulus $\{z \in \mathbb{C} : 1 < |z| < 3\}$. This is an old ...
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### Function satisfying inequality has no root

Let $f$ be an entire function such that, for all $z \in \mathbb{C}$ with $|z| > 1$, $$|f'(z)| < \frac{|f(z)|}{|z|^2} < 1$$ Show that there is no $a \in \mathbb{C}$ such that $f(a) = 0$. ...
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### Show that … Has No Real Roots

When $f(x) = 3x-4$ and $g(x) = \frac{5}{3-x}$, Question 1: Find the value of x for which fg(x) = 5 Question 2: Show that the equation $f^{-1}(x) = g^{-1}(x)$ has no real roots. I understand that ...
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### How would you find the roots of $x^3-3x-1 = 0$

I'm not too sure how to tackle this problem. Supposedly, the roots of the equation are $2\cos\left(\frac {\pi}{9}\right),-2\cos\left(\frac {2\pi}{9}\right)$ and $-2\cos\left(\frac {4\pi}{9}\right)$ ...
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### How to solve a quintic polynomial equation?

I know that not all quintics are solvable. But how do I identify the class of solvable ones?
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### How can I imagine double/repeated root of a quadratic equation?

A quadratic equation such as $(x-2)^2=0$ has a repeated root $(2,2)$. A lot of things in math (equations, matrixes and so) can be nicely drawn on a $2D, 3D$ etc plane (with $x$-$y$ axis). I mean, I ...
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### How can I simplify $\frac{xy-\sqrt{\frac{(1-x^2)(1-y^2)}{(1+x^2)(1+y^2)}}}{1+xy\sqrt{\frac{(1-x^2)(1-y^2)}{(1+x^2)(1+y^2)}}}$? [closed]

Came across this on another math forum, can't seem to solve.
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### How would you find the exact roots of this equation?

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 ...
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### Finding an interval in which all the real roots of a polynomial lie

I'm making a program which uses simple bisection method to find the roots of a polynomial. For me to implement this method, I need a rough interval where it can be said with absolute certainty that ...
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### Why does degree determine the amount of zeros?

We just learned about complex numbers in my math class and I have a question. Why does the degree of a polynomial equal the amount of zeros it has? The degree of $f(x) = x^3 - x^2 + x - 1$ is $3$, ...
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### Solutions of an equation of degree $n>4$
I know that the Abell-Ruffini theorem prove that we cannot solve a general equation of degree $n>4$ with radicals. But I've read that quintic equations can be solved by means of elliptic modular ...
My polynomial is this ten term monster $P(x,y,z) = 6561 x^3+486 x^2+12 x+6561 y^3+1944 y^2+192 y+6561 z^3+6318 z^2+2028 z+223$ It's simplest form is \${1 \over 81} \left( (81x+2)^3 + (81y+8)^3 +(81z+...