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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 $x\in \mathbb{... 10answers 9k views Why can a quadratic equation have only 2 roots? It is commonly known that the quadratic equation$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 ... 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 ... 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, ... 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 $$e^... 3answers 2k views How do I prove that a polynomial F[x] of degree n has at most n roots It's a really basic question,in these days, I've been thinking why a polynomial p(x)\in F[x] (F a field) with degree n can have at most n roots. It seems easy to prove, but I've been trying to ... 3answers 3k views If a quadratic equation ax^2+bx+c=0 has more than two roots, then a=b=c=0 [closed] If a quadratic equation ax^2+bx+c=0 has more than two roots, then it is an identity i.e. it is true for all values of x and a=b=c=0. What is a proof of this? 2answers 3k views Roots of a polynomial in an integral domain Let R be a ring and f(X) \in R[x] be a non-constant polynomial. We know that the number of roots, of f(X) in R has no relation, to its degree if R is not commutative, or commutative but not ... 2answers 91 views Sum of the thirteenth power of the roots of given polynomial Find the sum of the thirteenth powers of the roots of x^{13} + x - 2\geq 0. Any solution for this question would be greatly appreciated. 2answers 70 views Is this a correct interpretation of the fundamental theorem of algebra? I tried reading the Wikipedia page but it's stated in terms of complex roots, and I don't really understand how that relates to the following proposition: if a real valued polynomial:$$\sum_{i=0}^n ... 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 ... 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 ... 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:...
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?
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 ...