# Questions tagged [roots]

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 the (arithmetic) tag. For questions about roots of Lie algebras, use the (lie-algebra) tag instead.

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### Can we say that $x=0$ is a double root of $f(x) = (e^x-1)(\ln( x+1))$?

Let $$f(x) = (e^x-1)(\ln( x+1))$$ So far , I've only seen examples of textbooks referencing repeated roots if we can write the function with linear factors raised to some natural exponent ( like if $a$...
58 views

### Roots of the polynomial $z^4+4z^3+6z^2-4z+4$ in the circle $|z+1|<1$.

I came across the following problem: Find the number of zeros of $P(z)=z^4+4z^3+6z^2-4z+4$ in the circle $|z+1|<1$. I know that I can't apply Rouche's Theorem directly since we don't have a ...
36 views

### Minimal distance between zero's $d = D(f(z)) = \inf_{i \neq j} |(z_i - z_j)| s.t. f(z_n) = 0$?

Let $f(z)$ be a transcendental entire function. Hence $f$ is not a polynomial. Assume $f(z)$ has infinitely many zero's $z_n$ $$f(z_n) = 0$$ Lets say that $f(z)$ is given by a taylor series. Im ...
35 views

### Is the fact that complex roots of polynomials with real coefficients occur in pairs of any use if we have a polynomial and know one real root only?

Consider the following theorem appearing in Linear Algebra Done Right 4.15 Polynomials with real coefficients have zeros in pairs. Suppose $p\in P(\mathbb{C})$ is a polynomial with real coefficients. ...
37 views

### Reverse Engineering Squared Squared [closed]

So i'm working on an encryption module and im having a wee bit of trouble. Lets say I have multiple numbers between 2 and 8. Now I square the numbers with each other. a * a * b * b * c * c * d * d = ...
96 views

### Permutating the coeffecients of equations with roots of unity.

I"ve noticed the roots of $$(x^k-1 =0) = (1-x^k=0)$$ This is simple enough since changing +/- signs gives you the additive inverse which is equal at zero. However this property can be also be ...
1 vote
93 views

### Find the sum of cube root of the roots of a cubic equation.

Question. Given roots $a,b,c$ of the cubic equation $x^3+3x^2+11x+1=0$, find $$s=\sqrt{a}+\sqrt{b}+\sqrt{c}.$$ Note that this polynomial is irreducible over $\Bbb Q$. Vieta's formula was my ...
25 views

### Find range of constant part of 4th degree polynomial

Suppose there is a 4th degree polynomial ax^4 + bx^3 +cx^2 + dx + e then what will be the range of e so that this equation will have four real roots. I thought about using differentiation to try it ...
1 vote
23 views

1 vote
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### Finding the 8th and 10th sum of a polynomial. roots of polynomial equations [closed]

The cubic equation has roots $$x^3-x+4=0$$ has roots α, β, Γ find the cubic equation that has the roots $α^2, β^2, Γ^2$ hence or otherwise determine the values of S6, S8 and S10 where Sk is the sum ...
We use $P_n$ to denote the vector space of complex polynomials of degree $\leq n$, and we write $$\mathcal{P}^{*}_n = \mathcal{P}_n \setminus \mathcal{P}_{n-1}$$ for the set of polynomials of exact ...