Is this answer for finding the roots of a polynomial rigorous enough?

Question

Consider the following elementary question and answer:

Q: Find all the real roots of $P(x)=x^2-x$.

A:

$x^2-x=0$

$x(x-1)=0$

$x=0$ or $x-1=0$

$x=0$ or $x=1$

$x=0,1$. End.

This is a typical answer given by elementary school students and teachers. However, I wonder whether this answer is actually correct or valid in the context of proofs, logic, and rigor. To me, this answer only proves "If $x^2-x=0$, then $x=0$ or $x=1$", and doesn't show the converse "If $x=0$ or $x=1$, then $x^2-x=0$". The first one is needed to show no other real roots exist, while the second is needed to show these are indeed roots, and both are needed to show that the set {0,1} contains all the roots.

Questions

1. Do the statements in the answer have an if, then or an if and only if relationship? i.e., does the answer mean "$x^2-x=0 \iff x=0$ or $x=1$" or "$x^2-x=0 \Rightarrow x=0$ or $x=1$"?

   • I know that the iff relationship is true for all the equations in the answer, but I would like to know whether it is what is meant in this case, given that the iff symbol is not written down between the equations.

2. Would the answer be correct in the eyes of a teacher who demands 100% rigor? Or would it be necessary to plug the values back into the polynomial to show the converse?

3. Would writing the iff symbol before each equation be needed, correct, or bad practice (in proofs)?

4. When a question says "$P(x)=C$. Solve for $x$.", is there a formal restatement of the question in symbolic logic?

Answer 1

You certainly have a wonderful eye for detail.

Yes, broadly speaking, unless we can (in essence) connect every piece of our root-finding process with an "if and only if" statement, there might be information lost: we might introduce roots that weren't there to begin with. (The easy example is with $x = 1$; square it, and suddenly $-1$ is a root, but $x=1 \;\;\not \!\!\!\!\iff x^2 = 1$.)

1. Do the statements in the answer have an if, then or an if and only if relationship? i.e., does the answer mean "$x-1=0 \iff x=1$" or "$x-1=0 \Rightarrow x=1$"?

It is an "if and only if" situation.

Suppose $x-1=0$. Add one to both sides to conclude that $x=1$.

Suppose $x=1$. Subtract one from both sides to conclude that $x-1=0$.

More broadly, if you "do the same thing" to both sides of an equation, and that operation is well-defined and injective (i.e. $f(x)=f(y) \implies x=y$; equivalently, $x \ne y \implies f(x) \ne f(y)$), then one can use "if and only if" as a sort of "logical connective".

2. Would the answer be correct in the eyes of a teacher than demands 100% rigor? Or would it need to plug the values back into the polynomial to show the converse?

Now this is a question that almost merits two separate answers.

In the eyes of the person who demands perfect rigor, no, the proof in the OP might be sufficient and might not be. They might, in particular, expect every logical equivalence to be fully justified.

On the other hand, we are talking about a teacher. At some point in our mathematical education, we get a feel for statements that we can take for granted. I don't feel particularly obligated to mention explicit use of the quadratic formula or show the intermediate steps in my calculations, now that I'm well beyond calculus, unless I'm in some strange context or using different number systems. Even in my most rigorous math classes, such items could be taken for granted, for brevity's sake -- the rigor was exercised more on new material than on old stuff, unless it was particularly enlightening or important for said old material.

In that sense, even in a class with a high level of rigor (and assuming we're working over $\mathbb{C}$), I don't think a teacher would even look at $$x^2 - x = 0 \implies x \in \{0,1\}$$ as written with more than a passing glance. It's true, and you can certainly solve it.

This of course begs the third note -- you should ask your hypothetical teacher the level of rigor demanded to you by the work, and which items can be assumed and which cannot. Some expectations can probably be gleaned from class notes as well.

3. Would writing the iff symbol before each equation be needed, correct, or bad practice (in proofs)?

Entirely fine; I do it often myself (when the intermediate steps are simple enough to merit an explanation limited to a small line of text), and have never been criticized for it. I'm not sure what else you could reasonably do if you want to show your work (aside from explaining each step line-by-line, but I feel like that would break the flow a lot).

4. When a question says "P(x)=C. Solve for x.", is there a formal restatement of the question in symbolic logic?

No, because logical statements are just that -- statements, declarative in nature. They don't do imperatives like "solve for $x$" very well.

Answer 2

This is a typical answer given by elementary school students and teachers.

Would writing the iff symbol before each equation be needed, correct, or bad practice (in proofs)?

Given that the iff symbol is not written down between the equations, which is meant in this case?

Would the answer be correct in the eyes of a teacher who demands 100% rigor?

While inserting ⟺ symbols certainly clarifies the given presentation (alternatively, as suggested by Lutz in the above comments, ‘It is sometimes supremely helpful to add text to a solution: you could write "In the following calculation all transformations are equivalencies" and leave the lines as they are.’), unless I have previously explicitly communicated to the class their requirement, I would award it full credit and just treat the ⟺ symbols as meant to be tacit.

The inclusion/omission of the ⟺ symbols isn't necessarily about rigour; after all, how one writes a piece of mathematics, like with any informal communication, depends on their intended reader. By ‘informal communication’, all I mean is that mathematics is neither source code nor formal logic.

Or would it be necessary to plug the values back into the polynomial to show the converse?

If the ⟺ symbols are indeed at least tacit, then plugging back the solution values (that is, the converse argument direction) is already implicit in the presentation.

However, when introducing this topic, I think it's too idealistic (see below) to insist on the ⟺ argument, in which case it is necessary to plug the solution values back into the polynomial. Whether this needs to be made explicit is again a matter of mutual agreement between the instructor and class. (Sorry that this is not a very satisfying answer!)

Quoting myself here and here:

In practice, because trying to ensure invertibility at every step is tedious and prone to carelessness, when solving equations, it is best to just work in a forward direction like this $$\sqrt{x+3}=x+1 \color{#C00}\implies x+3=(x+1)^2 \color{#C00}\implies x=-2\;\text{or}\;1,$$ then plugging the candidate solutions into the given equation to sift out any extraneous ones.

Most people (and in particular the middle-schoolers under discussion) will understandably soon forget about the converse direction anyway, and it is better to help develop their intuition for spotting the typical steps that potentially produce extraneous solutions.

Answer 3

Would writing the iff symbol before each equation be needed, correct, or bad practice (in proofs)?

• Needed: almost certainly
• Correct: Definitely
• Bad practice: I find it difficult to think of a situation where it would be bad practice.

I try to follow the conventions in On the shape of mathematical arguments by A. J. M. van Gasteren. She advocated this shape for (many) proofs: $$P$$ $$= \{hint\;why\;[P\iff Q]\}$$ $$Q$$ $$\implies \{hint\;why\;[Q\implies R]\}$$ $$R$$

When a question says "P(x)=C. Solve for x.", is there a formal restatement of the question in symbolic logic?

That's a hard one, as I don't think symbolic logic does active verbs. IMHO the best we can do is to write in English (or French, German, Te Reo, etc) $\text{find x such that } P(x)=C$