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Fundamental Theorem
Turning Points
Let's talk about turning points. In the introduction, we noticed that the number of turns in the polynomial graphs we worked with were usually one less than the degree of the polynomial.
This is not always true. Remember when we did just x2, x4, etc. And the graph looked similar to a parabola with a wider base?
What about the odd powers, like x5 or x7? They all looked very similar to the cubic graph, just different around the origin.
Other polynomial equations as well may not look like they have the turning points unless you zoom in quite a bit, but as a general rule, when working with polynomials we expect there to be one less turning point than the power.
x-Intercepts
Now let's talk about the x-intercepts.
In the introduction, we had one third degree equation that had one x-intercept instead of three.
See the graph to the right. It looks like the others, but the turns did not reach the x-axis. This is because it has imaginary roots here. The imaginary roots always come in pairs.
The x-intercepts are the zeros of the function. The imaginary numbers are also called the zeros. The graph of a polynomial function of degree n has at most n - 1 turning points and at most n x-intercepts.
If all of this is considered, this equation would then fit the pattern and leads us to the Fundamental Theorem of Algebra.
Theorem
The Fundamental Theorem of Algebra:
Every polynomial of degree n has n zeros when considering multiple roots and imaginary numbers.
I use the word expect.
If the polynomial is third degree, I expect to find three zeros and/or three x-intercepts.
I might not find them all. Some may be imaginary or sometimes there could be repeated roots, but I am going to look for three.
If the polynomial is fifth degree, I am going to look for five.
What Do You Expect?
How many zeros/x-intercepts would you expect to find (would there be at most) for the following equations?
P(x) = 3x5 + 2x3 − x2 + 5 = 5
P(x) = 2x6 + 5x7 − x8 + 1 = 8
P(x) = −5x2 + 2x6 − x3 − 8 = 6
How many turning points do you expect to find (would there be at most) for the equations above?
4, 7, 5
More About Theorem
We need to discuss one more thing and that is to show that the fundamental theorem of algebra holds true for quadratics.
Think back to the quadratic formula for just a minute.
Because there is a ± in the formula, we will always get two answers. They might be the same thing, a double, or they might be imaginary, but there will always be two using this formula.
Now think back to what you've learned about the discriminant.
Discriminants (b2 − 4ac > 0 or D > 0)
Recall that the portion of the quadratic formula under the radical b2 − 4ac is called the discriminant. The discriminant may have 3 different values. b2 − 4ac > 0 or D > 0 (with D representing the discriminant)
When the value of the discriminant is larger than zero, the value is a positive number. This tells us that the quadratic will have 2 real roots which also means we have two x-intercepts.
Discriminants (b2 − 4ac = 0 or D = 0)
Our second possibility for the value of the discriminant is:
- b2 − 4ac = 0, or
- D = 0
This means that the quadratic function will have a double root since we will have 0 as a value under the radical. If we add or subtract zero to or from a value, we get the same number. This means that we will also have one x-intercept.
Discriminants (b2 − 4ac < 0 or D < 0)
The last possibility for the value of the discriminant is:
- b2 − 4ac < 0, or
- D < 0
When D < 0, the value under the radical in the quadratic formula is negative. We remember that taking the square root of a negative number give us an imaginary number. A quadratic equation with a negative discriminant has 2 imaginary roots and zero x-intercepts.
End Behavior
End Behavior of Polynomial Functions Degree 0 - 2
The general shapes of different polynomial functions are shown below with the maximum number of turning points and x-intercepts.
End Behavior of Polynomial Functions Degree 3 - 5
The general shapes of different polynomial functions are shown below with the maximum number of turning points and x-intercepts.
Even Degree Polynomials
We are going to look at the end behavior of polynomials. If you will notice with the even degree polynomials, the graphs start and end in the same direction. The graphs both start up and end up because the leading coefficients are positive. If the leading coefficients were negative, the graphs would start down and end down.
Odd Degree Polynomials
If you will notice with the odd degree polynomials, the graph starts and ends in opposite directions. The graphs both start down and end up because the leading coefficients are positive. If the leading coefficients were negative, the graphs would start up and end down.
End Behavior of Polynomial Functions Summary
Let's summarize the information from the previous slides:
| Even Degree Polynomials | Odd Degree Polynomials | |
|---|---|---|
| Graph Starts and Ends | In the same direction | In opposite directions |
| Leading Coefficient is Positive |
|
|
| Leading Coefficient is Negative |
|
|
Positive Even Degree Polynomials
Starts up and ends up.
As x goes to (can be denoted by an arrow) negative infinity, where is f(x) going?
The graph is going up, so f(x) or the y-values of the graph are going to positive infinity.
As x goes to (can be denoted by an arrow) positive infinity, where is f(x) going?
The graph is going up, so f(x) or the y-values of the graph are going to positive infinity.
You will write your answer for the end behavior in the following format:
As x --> - ∞, f (x) --> ∞
As x --> ∞, f (x) --> ∞
Negative Even Degree Polynomials
Starts down and ends down
As x goes to (can be denoted by -->) negative infinity, where is f(x) going?
The graph is going down, so f(x) or the y-values of the graph are going to negative infinity.
As x goes to (can be denoted by -->) positive infinity, where is f(x) going?
The graph is going down, so f(x) or the y-values of the graph are going to negative infinity.
You will write your answer for the end behavior in the following format:
As x --> - ∞, f (x) --> - ∞
As x --> ∞, f (x) --> - ∞
Positive Odd Degree Polynomials
Starts down and ends up
As x goes to (can be denoted by -->) negative infinity, where is f(x) going?
The graph is going down, so f(x) or the y-values of the graph are going to negative infinity.
As x goes to (can be denoted by -->) positive infinity, where is f(x) going?
The graph is going down, so f(x) or the y-values of the graph are going to positive infinity.
You will write your answer for the end behavior in the following format:
As x --> - ∞, f (x) --> - ∞
As x --> ∞, f (x) --> ∞
Negative Odd Degree Polynomials
Starts up and ends down
As x goes to (can be denoted by -->) negative infinity, where is f(x) going?
The graph is going down, so f(x) or the y-values of the graph are going to positive infinity.
As x goes to (can be denoted by -->) positive infinity, where is f(x) going?
The graph is going down, so f(x) or the y-values of the graph are going to negative infinity.
You will write your answer for the end behavior in the following format:
As x --> - ∞, f (x) --> ∞
As x --> ∞, f (x) --> - ∞
Even/Odd Degree Polynomials Summary
Let's summarize the information from the previous slides.
| Even Degree Polynomials | Odd Degree Polynomials | |
|---|---|---|
| Graph Starts and Ends | In the same direction | In opposite directions |
| Leading Coefficient is Positive |
|
|
| Leading Coefficient is Negative |
|
|
Example #1
Watch Describe End Behavior of a Polynomial Function Given Its Equation.
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Example #2
Watch Describe End Behavior of a Polynomial Function Given Its Equation.
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