Linked Questions

0
votes
2answers
27 views

Why is a polynomial with infinite zeropoints the zeropolymomial? [duplicate]

This was given us as a fact, but why is this true? The zeropolynomial is the polynomial where all the coefficients are equal to $0$ if $R(x)$ is a polynomial over $\mathbb{C}$ and every $x\in \mathbb{...
40
votes
10answers
9k views

Why can a quadratic equation have only 2 roots?

It is commonly known that the quadratic equation $ax^2+bx+c=0$ has two solutions given by: $$x = \frac{-b\pm \sqrt{b^2-4ac}}{2a}$$ But how do I prove that another root couldn't exist? I think ...
3
votes
4answers
999 views

Polynomial equal to polynomial of lower degree

I am studying Linear Algebra Done Right, chapter 2 problem 6 states: Prove that the real vector space consisting of all continuous real valued functions on the interval $[0,1]$ is infinite ...
7
votes
4answers
2k views

Prove that a polynomial of degree $d$ has at most $d$ roots (without induction)

Let $p(x)$ be a non-zero polynomial in $F[x]$, $F$ a field, of degree $d$. Then $p(x)$ has at most $d$ distinct roots in $F$. Is it possible to prove this without using induction on degree? If so, ...
3
votes
3answers
2k views

Prove that $e^x, xe^x,$ and $x^2e^x$ are linearly independent over $\mathbb{R}$

Question: Prove that $e^x, xe^x,$ and $x^2e^x$ are linearly independent over $\mathbb{R}$. Generally we proceed by setting up the equation $$a_1e^x + a_2xe^x+a_3x^2e^x=0_f,$$ which simplifies to $$e^...
1
vote
3answers
2k views

How do I prove that a polynomial F[x] of degree n has at most n roots

It's a really basic question,in these days, I've been thinking why a polynomial $p(x)\in F[x]$ ($F$ a field) with degree $n$ can have at most n roots. It seems easy to prove, but I've been trying to ...
0
votes
3answers
3k views

If a quadratic equation $ax^2+bx+c=0$ has more than two roots, then $a=b=c=0$ [closed]

If a quadratic equation $ax^2+bx+c=0$ has more than two roots, then it is an identity i.e. it is true for all values of $x$ and $a=b=c=0$. What is a proof of this?
5
votes
2answers
3k views

Roots of a polynomial in an integral domain

Let $R$ be a ring and $f(X) \in R[x]$ be a non-constant polynomial. We know that the number of roots, of $f(X)$ in $R$ has no relation, to its degree if $R$ is not commutative, or commutative but not ...
2
votes
2answers
91 views

Sum of the thirteenth power of the roots of given polynomial

Find the sum of the thirteenth powers of the roots of $x^{13} + x - 2\geq 0$. Any solution for this question would be greatly appreciated.
2
votes
2answers
70 views

Is this a correct interpretation of the fundamental theorem of algebra?

I tried reading the Wikipedia page but it's stated in terms of complex roots, and I don't really understand how that relates to the following proposition: if a real valued polynomial: $$\sum_{i=0}^n ...
1
vote
2answers
68 views

Finding trinity of complex numbers

I want to find all $x_1,x_2,x_3$ that satisfy these three equalities: $$x_1+x_2+x_3=6$$ $$x_1^2+x_2^2+x_3^2=14$$ $$x_1^3+x_2^3+x_3^3=36$$ So i don't know whether i should solve it using the ...
3
votes
1answer
157 views

Number of roots smaller than degree of polynomial

Let $R$ be a commutative ring and $f\in R[X]$ a polynomial with $f\neq 0$ and suppose $a_1,...,a_n\in R$ are roots of $f$ with $a_i-a_j\in R^*$ for all $i,j$ with $1\leq i<j\leq n$. How do I ...
0
votes
1answer
136 views

Why is the following true? Zero function and the zero polynomial

I am reading through a proof of the following by induction but stuck at a very early step. $k$ is an infinite field and let $f \in k[x_1,...,x_n]$. Then $f=0$ in $k[x_1,...,x_n]$ if and only if $f:...
0
votes
2answers
75 views

Connection between number of roots for a given polynomial and its degree

Why do we get $2$ solutions for a quadratic equation and $3$ solutions for a cubic equation and $4$ for biquadratic equation and so forth?
1
vote
2answers
91 views

Size of an ideal in a polynomial Ring

Let $F$ be a field and let $I = \{f(x) \in F[x]\mid f(a) = 0 ~~ \forall a \in F\}$. Prove that $I = \{0\}$ when $F$ is infinite. I have already shown that $I$ is an ideal and that $I$ is infinite ...

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