Linked Questions

1
vote
1answer
91 views

Can differentiation be defined in an algebraic way?

Is it possible to define the operation D of differentiation of real functions in an abstract way, as for example by the fundamental properties of the derivative: D(f+g) = D(f) + D(g) D(fg) = fD(g) ...
2
votes
1answer
473 views

The universal property of polynomial rings and universal identities

To verify some identity involving say two variables $x, y \in R$ for any commutative ring $R$ it suffices to verify this identity in $\mathbb{Z}[x,y]$. This is just the sort of trick that should be ...
3
votes
3answers
2k views

Determine the remainder when $f(x) = 3x^5 - 5x^2 + 4x + 1$ is divided by $(x-1)(x+2)$

This question arose while I was tutoring a student on the topic of the Remainder Theorem. Now, the Remainder Theorem tells us that when a polynomial $p(x)$ is divided by a linear factor $(x-a)$, the ...
56
votes
9answers
14k views

Has there ever been an application of dividing by $0$?

Regarding the expression $a/0$, according to Wikipedia: In ordinary arithmetic, the expression has no meaning, as there is no number which, multiplied by $0$, gives $a$ (assuming $a\not= 0$), and ...
3
votes
2answers
450 views

gcd of $x$ and $2$ in $\mathbb Z[x]$

In $\mathbb Z[x]$, $x$ and $2$ have gcd $1$. But they cannot be expressed as the linear combination of two polynomials. Then assuming that $1=2f(x)+xg(x)$ we are supposed to arrive at a ...
3
votes
1answer
214 views

Why is modular inverse notation so ambiguous?

Consider $$\frac{a}{a}\pmod a,\ \ \ a\in\mathbb Z\setminus\{-1,0,1\}$$ There are two cases: $1)$ $\frac{a}{a}$ is the notation for the real number $1$. Then the expression is equivalent to $1$ ...
2
votes
5answers
177 views

How to prove that $(x-1)^2$ is a factor of $x^4 - ax^2 + (2a-4)x + (3-a)$ for $a\in\mathbb R$?

Let $a \in R$. Verify that $(x − 1)^2$ is a factor of $$p(x) = x^4 − ax^2 + (2a − 4)x + (3 − a)$$ How can I solve this question?
1
vote
3answers
91 views

Very simple function question

Formally speaking, can $\frac{x}{x}$ be defined for $x=0$? Normally this division turns to be 1 but we also know that we can only divide it when the denominator is different from zero. In this case, ...
24
votes
5answers
2k views

Why are polynomials defined to be “formal”?

Despite the fact that $\forall n, n^3 + 2n \equiv 0 \pmod 3$, I understand that $n^3 + 2n$ (considered as a polynomial with coefficients in $\mathbb Z/3\mathbb Z$) is not equal to the zero polynomial. ...
3
votes
3answers
566 views

How to prove that if $a\equiv b \pmod{kn}$ then $a^k\equiv b^k \pmod{k^2n}$

What I have done is this: $a\equiv b \pmod{2n}$, $a=b+c\times2n$, for some $c$, $a^2=b^2+2b\times c\times2n+c^2\times2^2n^2$, $a^2-b^2=(b\times c+c^2n)\times4n$, then $a^2\equiv b^2\pmod{2^2n}$. ...
25
votes
4answers
8k views

Sylvester's determinant identity

Sylvester's determinant identity states that if $A$ and $B$ are matrices of sizes $m\times n$ and $n\times m$, then $$ \det(I_m+AB) = \det(I_n+BA)$$ where $I_m$ and $I_n$ denote the $m \times m$ ...