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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 ...