11 questions linked to/from cancelling out before evaluation of variable
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### 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 ...
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### 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 $m \times m$ ...
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### 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. ...
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### 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 $a^2\equiv b^2\pmod{2^2n}$. ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 $1$ ...
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### 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, ...
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### 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) ...