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When searching here to find methods for solving quartic polynomials, I came across a question where one of the solutions (at the very end) mentions that the equating coefficients method can fail. https://math.stackexchange.com/a/57688

The example given is $$x^4−x−1=0$$ and indeed when equating coefficients of $$x^4−x−1=(x^2+bx+c)(x^2+Bx+C)$$ the resulting system is inconsistent.

I've not been able to find a formal statement or proof of the equating the coefficients theorem to try to figure out if some assumption has been violated or why it doesn't work in this case.

I don't have the background yet to understand an explanation based on the Galois groups that are briefly mentioned in the link, so if someone could find another way to explain it, that would be appreciated. An additional explanation based on Galois groups may also help someone else though.

https://en.wikipedia.org/wiki/Equating_coefficients doesn't mention anything about where it might fail.

Thank you.

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T. M.
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1 Answers1

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(If you haven't heard of universal quantification, have a careful read of my answer here).

Over the real numbers, equating coefficients always works. For instance, the following is a genuine theorem of mathematics:

Theorem. Let $a,b,c$ and $a',b',c'$ denote real numbers. Then

$$\left[\left( \mathop{\forall}_{x\in \mathbb{R}} \right) ax^2+bx+c=a'x^2+b'x+c'\right] \rightarrow (a=a') \wedge (b=b') \wedge (c=c')$$

This never fails to work.

What can fail is the attempt write a polynomial as a product of polynomials of lower degree. For instance, let $a$ and $b$ denote real numbers. Then the statement $$\left(\mathop{\forall}_{x\in \mathbb{R}} \right) x^2+1 = (x-a)(x-b),$$ if assumed, implies a contradiction.

(The complex numbers solve this particular problem, but even over the complex numbers, many bivariate degree $2$ polynomials cannot be written as a product of degree $1$ polynomials.)

I'm not quite sure why you think that if $a,b$ and $A,B$ are real numbers, then the statement $$\left(\mathop{\forall}_{x\in \mathbb{R}}\right) x^4−x−1=(x^2+bx+c)(x^2+Bx+C),$$ if assumed, leads to a contradiction. In fact, I was under the impression that this doesn't lead to a contradiction! Can you offer a proof of your claim?

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goblin GONE
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  • I should add that I think its completely shameful that most schools teach the idea of "equating the coefficients" without ever teaching universal quantification. You cannot understand the former without first understanding the latter. – goblin GONE Feb 14 '16 at 18:51
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    Also, this can fail over finite fields, where polynomials of different degrees can have the same outputs! – Spooky Feb 14 '16 at 18:53
  • @Spooky, good point. I've edited the offending sentence: it now reads "over the real numbers, equating coefficients always works." – goblin GONE Feb 14 '16 at 18:54
  • And in fact, any infinite field :) – Spooky Feb 14 '16 at 18:56
  • @Spooky, of course. But all my universal quantifiers are probably intimidating enough. I didn't want to scare off the OP completely; just give them a slight push towards clarity of thought. – goblin GONE Feb 14 '16 at 18:58
  • I definitely didn't assume $${x\in \mathbb{R}}$$ since solutions for even quadratics aren't always in $$\mathbb{R}$$. What I looked at was – T. M. Feb 14 '16 at 19:05
  • @T.M., equating of coefficients also works over the complex numbers. Its only when you start looking into finite fields that you need to worry about it. – goblin GONE Feb 14 '16 at 19:08
  • Thank you, I think I can see my error now. I took the linked post too literally when it said "this method sometimes fails." and a mistake in solving the system of equations that resulted seemed to confirm that failure. Letting $x^4-x-1=(x^2+bx+c)(x^2+dx+e)$ and equating I got the system $\left{b+d=0, c+bd+e=0, cd+eb=-1, ec=-1\right}$, and not knowing how to solve that readily, I erroneously assumed e=1 and c=-1. Solving that system is a different question, but now I know that equating the coefficients is essentially as complicated as other methods of solving the cuartic. – T. M. Feb 15 '16 at 01:21