Polynomial functions

rachelmaddie

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I just need my work checked please #6
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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 5 points, so there are 5 real zeros.
 
Can you identify the five points of intersection? Visually, I'm seeing only four. What's up with that?
 
Is that consistent with your determination of odd degree? Think it through.
 
As you stated, yes the graph represents an odd-degree polynomial function.
f(x) —> - ∞ if x —> - ∞
f(x) —> + ∞ if x —> +∞


This meaning, the function f(x) must have an odd number of roots ( where the graph of a function intersects the x-axis )
The graph of the function intersects the x-axis 5 times ( it touches the origin so it has the repeated root of x = 0 ),
the total number of real solutions of f(x) is 5.

Please correct me if I'm wrong.
 
As you stated, yes the graph represents an odd-degree polynomial function.
f(x) —> - ∞ if x —> - ∞
f(x) —> + ∞ if x —> +∞


This meaning, the function f(x) must have an odd number of roots ( where the graph of a function intersects the x-axis )
The graph of the function intersects the x-axis 5 times ( it touches the origin so it has the repeated root of x = 0 ),
the total number of real solutions of f(x) is 5.

Please correct me if I'm wrong.
Which means there are 5 real zeros?
 
It's still only four intersections. Just because you count it multiple times doesn't mean it intersects multiple times. Some distinguish "cross" and "bounce". Are they both "intersections"?
 
Which means there are 5 real zeros?
Mostly correct. There are an odd number, counting multiplicities. Those last two words are important.

Five is the MINIMUM number. Could be 7 or 9 or anything else odd greater than that.
 
There are 4 intersections. Just look at them and count them.
There are an odd number of zeros, counting multiplicities.
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 5 points, so there are an odd number of zeros, counting multiplicities.

Is this correct?
 
Still no. How many intersections are there? Multiplicities do NOT change the number of intersections.

[math]y = x^{37}[/math] has ONE (1) intersection with the x-axis. It has 37 zeros, counting multiplicities.
 
Still no. How many intersections are there? Multiplicities do NOT change the number of intersections.

[math]y = x^{37}[/math] has ONE (1) intersection with the x-axis. It has 37 zeros, counting multiplicities.
Number of real zeros is 37?
 
As you stated, yes the graph represents an odd-degree polynomial function.
f(x) —> - ∞ if x —> - ∞
f(x) —> + ∞ if x —> +∞


This meaning, the function f(x) must have an odd number of roots ( where the graph of a function intersects the x-axis )
The graph of the function intersects the x-axis 5 times ( it touches the origin so it has the repeated root of x = 0 ),
the total number of real solutions of f(x) is 5.

Please correct me if I'm wrong.
I’m confused because of this comment
 
I’m confused because of this comment
It's just an example. It's not the problem you're working on.

The point is:
  • Intersections do NOT come with multiplicities. They are very visual.
  • Zeros CAN come with multiplicities.
 
Still no. How many intersections are there? Multiplicities do NOT change the number of intersections.

[math]y = x^{37}[/math] has ONE (1) intersection with the x-axis. It has 37 zeros, counting multiplicities.
So this would be the correct answer?
 
My example? No. Please read from the beginning and provide the complete response. There will not be a '37' in it.
This was my response but I’m still not understanding what I did wrong here..
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 5 points, so there are 5 real zeros.
 
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