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### 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 ...
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### 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......
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### 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 ...
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### 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 ...
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### 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) = ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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[...