To avoid confusion, it bears reminding to people unfamiliar with the words that, in math, ‘real’ doesn’t mean ‘actual’, just as ‘imaginary’ doesn’t mean ‘not actual’.
“Imaginary numbers” are every bit as real[sup]1[/sup] as “real numbers[sup]2[/sup]”.
[sup]1[/sup]vernacular sense of the word
[sup]2[/sup]math jargon sense of the term
In our standard number system, the number zero is unsigned. You can come up with your own system that includes signed 0s, but it won’t mean much in the “standard” (whatever “standard” means these days) math you might see in school.
Google will now plots functions as well. Put x^2 + 3x - 18 into Google and it will produce this plot complete with a slider to show you the evaluation at each point. The question is asking for where it’s equal to zero. The only downside to using Google is that they will begin to profile you as a nerd.
This is derailing quite a bit from the original question, but just to clear this up: any polynomial of degree n (i.e. x[sup]n[/sup] is the highest power of x present) can be uniquely rewritten in the form
The values a[sub]1[/sub], a[sub]2[/sub], … , a[sub]n[/sub] are the “roots” of the polynomial (and are not necessarily real numbers.) Now for some polynomials, two or more of these roots (the values of a[sub]1[/sub], a[sub]2[/sub], … , a[sub]n[/sub]) will be equal. For example, consider the polynomial x[sup]2[/sup] - 2x + 1. This can be rewritten as (x - 1)(x - 1); so a[sub]1[/sub] = a[sub]2[/sub] = 1, and we say that 1 is a double root of this polynomial (or, if you prefer, that “the roots are 1 and 1”. It’s not as accurate to say that “the roots are 1 and 1 and 1”, because x[sup]2[/sup] - 2x + 1 is not the same polynomial as (x - 1)(x - 1)(x - 1).
Oh, and to use your example: x[sup]2[/sup] = (x - 0)(x - 0), so zero is a double root of the polynomial (rather than a triple or quadruple or quintuple root.)
No, it really does make sense to say “the solutions are 0 and 0.” Another way to put it is to say that 0 is a root of multiplicity 2. See MikeS’s explanation. For reasons that are not immediately obvious, but which become more and more obvious the more you study polynomials, algebra, etc., it makes sense to say that a polynomial equation of degree 2 (or, more generally, n) has exactly 2 (or n) solutions, counting multiplicity (which means that some of those solutions may be repeated—they may occur more than once).
Yeah, like any other their other gadgets, like calculator results. Be sure you’re phrasing it as a function and not an equation. There’s a tutorial here.
…which is what I’m imagining in my head. Johnny’s example of, say, 1, 1, and 3 would have an inflection point at x=1. I’m calling that the same root. Now if you’re telling me that a root is defined as a[sub]1[/sub] in (x-a[sub]1[/sub]), then I guess I’ll take your word for it, but it doesn’t make much sense to me to count the same point twice, and Wiki doesn’t seem to agree with that definition.
Even if a root has multiplicities, it’s still one root.
You are misusing the term “inflection point” here. In standard mathematical terminology, a root is an inflection point just in case it has multiplicity 3 or greater. For example, x = 0 is not an inflection point of x^2, but it is an inflection point of x^3.
For some purposes, “root” might as well just mean “point where the function hits 0”. But for other purposes, what is most interesting is “point around which the function behaves the way x^n behaves around x = 0, counted n many times”, and the term “root” (counted with multiplicity) is standardly used for this latter, more informative sense as well.
Well, what does “multiplicity” mean but “counted more than once”?
That’s because you didn’t put quotes around it, and furthermore, Google doesn’t care about the ^, +, or - for web search. Anything with x, 2, 3x, and 18 somewhere on the page would’ve come up.
Right. To invoke google’s graphing, it needs to be 1) a function, not an equation (no equals sign) and 2) no quotes around said function. Otherwise, it works great.
That’s right, if there is a root of “multiplicity n”, then:
– for odd n, the graph crosses the x-axis; for odd n >= 3, that would be an inflection point.
– for even n, the graph is tangent to the x-axis but doesn’t cross it; in this case you have a relative min or max there.
That is for real roots only, of course. For the unreal roots the graph doesn’t come near the x-axis. (I keep wondering what it would look like to graph an equation y=f(x) in three dimensions, using one dimension for real x inputs and the other two for complex y outputs. Or even graph in 4 dimensions, to show real and unreal values for both x and y.) ETA: I meant the other way around. Imaginary x inputs with real y outputs.
ETA: Note to OP: How ya doing with this discussion? Are you following all the chatter about multiplicities and inflection points and imaginary roots and all that?
