It is known that, in the universe of complex numbers, the only root of the equation $x^2 - 2x + 1 = 0$ is $1$. Could we say that the equation has two equal real roots? Or should we say that the equation has one real root with multiplicity 2?
A root? Or two roots?
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algebra-precalculus
polynomials
terminology
roots
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14I was under the impression that those are just two ways of saying the same thing. – 2010-11-15
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0There's a reason why the statement of the Fundamental theorem of Algebra has the phrase "counting multiplicity" or something equivalent in its statement. One also hears the word "coalesce" in some contexts. – 2010-11-15
4 Answers
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"Two equal roots" and "a root of multiplicity (at least) two" mean the same thing.
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0Why *at least*? Does it mean that two equal roots and a root of multiplicity 3 (or 4, 5 etc) mean the same thing? – 2012-05-14
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2@Dilawar: Because language is inexact; if I had a root of multiplicity 3, we would have "three equal roots", but if you have three things that are equal, then you also have *two* things that are equal. Of course, with a quadratic, multiplicity cannot exceed 2 in the first place. – 2012-05-14
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If you want to be totally precise, you have to say that it has one real root with multiplicity 2. But "two equal real roots" is almost universally understood as a shortcut for saying that.
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Both are correct, as well as saying that the root is degenerate (with multiplicity of two).
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Yes, all these answers are correct. I am not allowed to upvote them since I'm not registered, but I do know that the parabola(or quadratic function) you have there touches the graph on the x-axis in precisely 1 point. That is the reason why it is the solutions to your quadratic will be a rational, real, and equal root.
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0Consider registering for an account so you can upvote! :) – 2012-05-14