Have you ever wondered where you come from? Who your great-great-grandparents were? Well, polynomial functions wonder about their roots, just as we do! In mathematics, a polynomial function is an expression of finite length constructed from variable and constants. Within this function are roots, which determine the value(s) of x of said function.
Let's get started!
A zero, or root, of a polynomial function is the value of x such that f(x)=0. Zeros are also called the
x-intercepts of a polynomial, or points where the graph touches the x-axis.
The multiplicity of a member of a multi-set is is the number of times it appears in the set.
Example:
How many times does each of the following numbers occur in the given set of values?
a) x=2; x=2; x=2 ----------------------------------------------------> x=2 has a multiplicity of three.
b) x=4; x=5; x=5; x=4 ---------------------------------------------> x=4 has a multiplicity of two and
x=5 also has a multiplicity of two.
c) x=0; x=43 ----------------------------------------------------------> x=0 and x=43 both have a
multiplicity of one.
Application.
How does this apply to polynomial functions, you may ask? Well, remember way back when we were taught how to factor polynomials? Turns out, the more of the same members of x that there are, the more different the graph will look. By different, I mean whether the graph will pass through the given value of x, or if it will decide to bounce off of it.
Example:
The graph of the function f(x)=(x-2)^2 looks as follows:
Why? Let's take a look at f(x)=(x-2)^2.
If we remember the definition of a root, it states that a zero is a value at which f(x)=0.
With that said, let's set our f(x) equal to zero.
And this is what we get:
0=(x-2)^2
Which can also be written as:
0=(x-2)(x-2)
If we continue to find the values of x, we get:
x-2=0 and x-2=0
So…….
x=2 and x=2
Remember in our first example when we counted the number of times a specific value occurred in the given set of data? This is the same approach. We now have two value of x=2 ….. so a multiplicity of two. WOAH!
***Here is a standard rule to remember: if the multiplicity of a number of x is even, the graph will "bounce off" that value and continue its journey. If the multiplicity of a number of x is odd, then the graph will pass through the point and continue onto the next value.
Let's do another example…but more difficult:
f(x)=-(x+1)^3
This function includes a few points of discussion.
First, notice the exponent of 3. This shows us that the graph will pass through the x point..since 3 is an odd number.
Second, we have a nifty little negative sign at the beginning of the function. It does not affect the value of x, however, it does tell us the positioning of the ends of the graph.This negative sign tells us the end placements of the graph. Graphs are read left to right. If the second tail is facing the negative side of the x-axis, then the function is negative. If the second tail faces the positive side of the x-axis, then the function is positive.
Third, there is only one value of x.
So, we can paint a pretty vivid picture with this information. The graph will begin with a tail in the positive side of the x-axis, swoop down to the given value of x, pass through said value, and end in the negative side of the x-axis.
Was this conclusion correct?
Correct indeed!
Let's take a closer look at why that is. This time, I will solve it without commentary.
f(x)=-(x+1)^3
0=-(x+1)^3
0=-(x+1)(x+1)(x+1)
0=(x+1); 0=(x+1); 0=(x+1)
x=-1; x=-1; x=-1
Voila! Behold, a wonderfully complete polynomial function.
For further reference, please watch the following videos:
http://www.youtube.com/watch?v=DC-VNI6gGcs
The video above was quite informative. Ms. Stacy Reagan covers all of the points discussed above, and then some. Plus, she has a nice southern accent :)
alsooo……..
http://www.youtube.com/watch?v=K1Kv2p_Cmvc
This dude, Mr. Charlie, has a pretty good idea of what's what.
I hope this was semi-informative. :)
~Anastasiya Tsuker
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