Polynomial functions

This depends strongly on what you mean by "intersection". I see only three points where the graph actually crosses the x-axis. At x= 0 the graph goes down to the x-axis, touches it, then goes back up. If you count that as an intersection then there are 4 intersections. There are 5 (at least) zeros because x= 0 is a multiple (even multiplicity) zero.
 
HallsofIvy said:
This depends strongly on what you mean by "intersection". I see only three points where the graph actually crosses the x-axis. At x= 0 the graph goes down to the x-axis, touches it, then goes back up. If you count that as an intersection then there are 4 intersections. There are 5 (at least) zeros because x= 0 is a multiple (even multiplicity) zero.
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Describe the end behavior
f(x) —> - ∞ if x —> - ∞
f(x) —> + ∞ if x —> +∞
Determine whether it represents an odd-degree or even-degree function
Since the end behavior is in opposite directions, the graph represents an odd-degree polynomial function.

State the number of real zeros
The graph intersects the x-axis at 4 points, and there are 5 real zeros since x= 0 is a multiple.

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Rachel, intersections and zeroes just mean slightly different things.

There exists for any polynomial of degree n an equivalent expression that is the product of n polynomials of degree 1. Each of those polynomials of degree 1 determines a zero. So people say that a polynomial of degree n has n zeroes.

Let's see an easy example.

[MATH]x^2 - 6x + 9 = (x - 3)(x - 3).[/MATH]
We factored that quadratic into 2 linear terms so we say it has two zeroes. However, the two zeroes have the same value, namely 3, so there is only one intersection.

The way I remember it is that a polynomial of degree n always has n zeroes, but not every zero represents a real intersection. The number of zeroes determines the maximum number of intersectiions, but there may be fewer. The minimum number of intersections a polynomial of odd degree can have is 1; the minimum number of intersections a polynomial of even degree can have is zero.

Whenever we see an intersection that is a bounce off the x axis, that means two or more zeroes have the same value, and it is called a multiplicity. If you see a bounce that does not touch the x-axis, then there are zeroes that are not real numbers.

EDIT: I said that a factoring exists. That does not mean that there exists a way to find it exactly.
 
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rachelmaddie said:
Describe the end behavior
f(x) —> - ∞ if x —> - ∞
f(x) —> + ∞ if x —> +∞
Determine whether it represents an odd-degree or even-degree function
Since the end behavior is in opposite directions, the graph represents an odd-degree polynomial function.

State the number of real zeros
The graph intersects the x-axis at 4 points, and there are 5 real zeros since x= 0 is a multiple.

Like this?
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Again, almost- and probably what was intended! But since we don't know what the actual function is, it is possible that the multiple root at x=0 is NOT just a double root, a sextuple root, quadruple root, etc. We do know, since the graph goes down to y= 0 then back up, it is a root of even multiplicity. That is why TKHunny said, earlier, that "Five is the MINIMUM number. Could be 7 or 9 or anything else odd greater than that."

I suspect that, if you said all that, your teacher, while marking it "correct", would think "Show off!"
 
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HallsofIvy said:
Again, almost- and probably what was intended! But since we don't know what the actual function is, it is possible that the multiple root at x=0 is NOT just a double root, a sextuple root, quadruple root, etc. We do know, since the graph goes down to y= 0 then back up, it is a root of even multiplicity.
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So how would I word the second step?
 
rachelmaddie said:
So how would I word the second step?
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The point that Halls was making is that the question is badly worded. There is no way to tell how many real zeroes there are. There are four distinct real zeroes. And there are at least five real zeroes, one being duplicated, but we cannot tell how many times it is duplicated (other than it is an even number of times).
 
JeffM said:
The point that Halls was making is that the question is badly worded. There is no way to tell how many real zeroes there are. There are four distinct real zeroes. And there are at least five real zeroes, one being duplicated, but we cannot tell how many times it is duplicated (other than it is an even number of times).
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So how would I properly answer the question?
 
rachelmaddie said:
So how would I properly answer the question?
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Who can be sure? I would probably answer "there are four distinct real zeroes, and, if we assume that this function is of degree five, five real zeroes, but there is insufficient information to determine anything more except that the polynomial is of odd degree > 3 and has no complex roots." But that might be a bad answer if your teacher is stupid and just looks at the answer key.
 
JeffM said:
The point that Halls was making is that the question is badly worded. There is no way to tell how many real zeroes there are. There are four distinct real zeroes. And there are at least five real zeroes, one being duplicated, but we cannot tell how many times it is duplicated (other than it is an even number of times).
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Actually that was the point that TKHunny made before me.
 
HallsofIvy said:
Actually that was the point that TKHunny made before me.
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Yes he did, My point, however, is that the technical vocabulary truly is a bit confusing. Socratic questioning is not, in my opinion, useful for clarifying subtle distinctions among definitions.

The number of intersections of a polynomial with the x=axis equals the minimum number of distinct real roots. The degree determines the number of roots of all types and therefore the maximum number of real roots and the maximum number of intersections. Points of tangency indicate where there are duplicated real roots, but do not specify the number of duplicates. And when a student is faced with a problem that ignores those distinctions, it is legitimate for the student to be bemused. .

This is a question where our general refusal to give explicit answers shows why that refusal should not be universal. In fact I have no idea what answer her teacher will find acceptable. There is a stupid answer: 5 real roots. There are quite a few correct answers, including "exactly 4 distinct real roots," "at least 5 real roots with one duplicated an unknown but even number of times, and " five real roots if the polynomial is of degree 5."
 
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