Linked Questions

258
votes
21answers
31k views

Why can ALL quadratic equations be solved by the quadratic formula?

In algebra, all quadratic problems can be solved by using the quadratic formula. I read a couple of books, and they told me only HOW and WHEN to use this formula, but they don't tell me WHY I can use ...
39
votes
11answers
9k views

Why can a quadratic equation have only 2 roots?

It is commonly known that $ax^2+bx+c=0$ have two solutions $\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ but how to prove that another root couldn't exist? I think derivation of quadratic formula is not enough......
17
votes
3answers
5k views

How to solve polynomial equations in a field and/or in a ring?

I'm studying for my exam, and I stuck on solving polynomials in a field and/or in a ring. Let me give you some examples: (1) Solve equation $x^2+4x+3=0$ in field $\mathbb{Z}_5$, $\mathbb{Z}_8$ and in ...
4
votes
3answers
50k views

Square root of 1 [closed]

I know this is a dumb question but I can't get the answer to another follow up question, What is the square root of 1? If the square root of 1 is itself then why does other square root of number not ...
9
votes
5answers
5k views

Polynomial $p(x) = 0$ for all $x$ implies coefficients of polynomial are zero

I am curious why the following is true. The text I am reading is "An Introduction to Numerical Analysis" by Atkinson, 2nd edition, page 133, line 4. $p(x)$ is a polynomial of the form: $$ p(x) = ...
2
votes
5answers
4k views

Proving that an integral domain has at most two elements that satisfy the equation $x^2 = 1$.

I like to be thorough, but if you feel confident you can skip the first paragraph. Review: A ring is a set $R$ endowed with two operations of + and $\cdot$ such that $(G,+)$ is an additive abelian ...
4
votes
4answers
7k views

What is an Integral Domain?

Okay, so almost 3 months into my abstract algebra, we just started rings. I have a few questions. A "trivial ring" is a ring with only one element. So $R={0}$ is a trivial ring. Understandable. ...
1
vote
3answers
2k views

Intuitive understanding regarding the number of roots of a polynomial over a field and invertibility

In considering: A polynomial over a field of degree n has at most n roots. -- How does this make use of the stipulation "over a field" - especially with an eye toward inveritbility ? Is it to insure ...
2
votes
2answers
1k views

Equality of polynomials: formal vs. functional

Given two polynomials $A = \sum_{0\le k<n} a_k x^k$ and $B =\sum_{0\le k<n} b_k x^k$ of the same degree $n$, which are equal for all $x$, is it always true that $\ a_k = b_k\ $ for all $0\le k&...
5
votes
2answers
3k views

Show that if $ R $ is an integral domain then a polynomial in $ R[X] $ of degree $ d $ can have at most $ d $ roots

Show that if $ R $ is an integral domain then a polynomial in $ R[X] $ of degree $ d $ can have at most $ d $ roots. Thoughts so far: I feel like I might be missing something here. If $ R $ is an ...
0
votes
5answers
110 views

Can you help me to solve this $ x^2+x+1 \equiv 0 \pmod{13} $?

Can you help me to solve this $ x^2+x+1 \equiv 0 \pmod{13} $ ?
4
votes
2answers
338 views

Multiplicative Selfinverse in Fields

I assume there are only two multiplicative self inverse in each field with characteristice bigger than $2$ (the field is finite but I think it holds in general). In a field $F$ with $\operatorname{...
3
votes
2answers
1k views

Identically zero multivariate polynomial function

Let $p$ be a prime and let $F=\mathbb{Z}/p\mathbb{Z}$. Can a nonzero multivariate polynomial $f\in F[x_1,....,x_n]$ such that $\mathrm{deg}_if< p$ for all $i=1,\ldots,n$ be identically zero as a ...
6
votes
1answer
3k views

What is a non Trivial Square Root?

I need to understand the concept behind a non trivial square root. Also how to answer these two questions and how to get to the answer? Give a non-trivial square root of 30 Give a non-trivial square ...
3
votes
2answers
775 views

Extension of the factor theorem

Motivation If you don't care the least bit about motivation, scroll down. The following is a standard result in a first algebra course: Factor Theorem. Let $R$ be an integral domain and $p\in R[...

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