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7

Here's why you should take great care when considering $\pm$ as an operator. It's not unusual to see a sentence of the form We deduce that $A=\pm B$ and hence that $C=D\pm E$. This isn't simply saying that both ($A$ is either $B$ or $-B$) and (either $C=D-E$ or $C=D+E$). When two $\pm$'s appear in the same sentence it is implied that they are both to be ...


6

When solving a quadratic equation, $$ax^2+ bx+ c = 0$$ we use shorthand for the two solutions, to include both $$x_1 = \frac {-b +\sqrt{b^2-4ac}}{2a}$$ and $$x_2 = \frac{-b - \sqrt{b^2 -4ac}}{2a}$$ Hence, the shorthand, $$x_i = \frac{-b\pm \sqrt{b^2-4ac}}{2a}.$$ I.e., the solutions to $ax^2+bx+c = 0$ are given by $$x\in \left\{ \frac {-b +\sqrt{b^2-4ac}...


2

I do not believe that this is a concern that would surface, or be worth surfacing, in the courses named by this question's title. In my reading, the question is analogous to worrying about whether you can ask about e.g. the thousands place of $7521$: To do so assumes that the base ten representation of $7521$ is unique, or, in particular, that "thousands ...


13

I'm going to rewrite this answer to clarify what I think the issue is. I think the OP is imagining a different definition of the ring $k[x]$ than most answerers are. Here are two reasonable definitions: $k[x]$ is the ring of formal expressions of the form $\sum_{j=0}^{\infty} p_j x^j$ with $p_j \in k$ and we require that $p_j$ is $0$ for $j$ sufficiently ...


1

Soon after introducing polynomials, students learn to add and subtract polynomials. Notice that if $f(x)$ and $g(x)$ are polynomials with degrees $n$ and $m$ respectively, and if $f(x)-g(x)=0$, then $n=m$. It follows that the degree of a polynomial is well-defined. So the proof that degree is well-defined is not difficult at all. On the other hand, students ...


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