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Polynomial Functions of Higher Degree
Polynomial Functions:
Before we start looking at polynomials, we should know some common terminology.
Definition: A polynomial of degree n is a function of the form
P x ( ) = a xn n + an โ 1 xn โ^1 + ...+ a x 1 1 + a 0
where. The numbers are called the coefficients of the polynomial. The number is the constant coefficient or constant term. The number , the coefficient of the highest power is the leading coefficient , and the term is the leading term.
an โ 0 a 0 (^) , a 1 (^) , a 2 , ... , an a 0
n
a n a x n
Notice that a polynomial is usually written in descending powers of the variable, and the degree of a polynomial is the power of the leading term. For instance
P x ( ) = 4 x^3^ โ x^2 + 5
is a polynomial of degree 3. Also, if a polynomial consists of just a single term, such as
Q x ( ) = 7 x^4 , then it is called a monomial.
Graphs of Polynomials:
Polynomials of degree 0 are constant functions and polynomials of degree 1 are linear equations, whose graphs are both straight lines. Polynomials of degree 2 are quadratic equations, and their graphs are parabolas. As the degree of the polynomial increases beyond 2, the number of possible shapes the graph can be increases. However, the graph of a polynomial function is always a smooth continuous curve (no breaks, gaps, or sharp corners).
Monomials of the form P(x) = xn^ are the simplest polynomials.
As the figure suggest, the graph of P(x) = xn^ has the same general shape as y = x^2 when n is even, and the same general shape as y = x^3 when n is odd. However, as the degree n becomes larger, the graphs become flatter around the origin and steeper elsewhere.
Power Functions:
Another general type of function is called a power function. This is a function of the form f(x) = xn^ , where n is any positive number. (If n is negative, we call it a reciprocal function, which will be considered later.) It is called a power function because the independent variable ( x in this case) is raised to a power.
The domain and range for this function are both the set of all real numbers. It would be advantageous for you to graph the power functions for n = 5, 7, etc. You will see that the graphs all have a similar shape.
The graphs of the functions g ( x ) = x^2 , g ( x ) = x^4 , g ( x ) = x^6 (i.e. n is an even number) will all give the graph of a parabola. This would also be useful exercise to graph these functions.
What will the domain and range be when n is an even number?
Hint: Will the value of g(x) ever be negative?
Example 1: Sketch the graph of the function P(x) = -x^3 + 2 by transforming the graph of an appropriate function of the form y = xn^. Indicate all x - and y -intercepts on the graph.
Solution: Based on the transformation techniques, we know the graph of P(x) = -x^3 + 2 is the reflection of the graph of y = x^3 in the x -axis, shifted vertically up 2 units. Thus,
Example 2: Determine the end behavior of the polynomial Q x ( ) = 6 x^6 โ 4 x^4 + 2 x โ 3.
Solution: Since Q has even degree and positive leading coefficient, it has the following end behavior:
y โ โ as x โ โ and y โ โ as x โ โโ
Using Zeros to Graph Polynomials:
Definition: If is a polynomial and c is a number such that , then we say that c is a zero of P. The following are equivalent ways of saying the same thing.
P P c ( ) = 0
- c is a zero of P
2. x = c is a root of the equation P x ( ) = 0
3. x โ c is a factor of P x ( )
When graphing a polynomial, we want to find the roots of the polynomial equation
. To do this, we factor the polynomial and then use the Zero-Product Property.
Remember that if , then the graph of
P x ( ) = 0
P c ( ) = 0 y = P x ( ) has an x -intercept at x = c , so
the x -intercepts of the graph are the zeros of the function.
Example 3: Find the zeros of the polynomial R x ( ) = x^2 โ 7 x + 12.
Solution:
Step 1: First we must factor R to get
R x ( ) = ( x โ 4 ) ( x โ 3 )
Step 2: Since x โ 4 is a factor of R x ( ) = x^2^ โ 7 x + 12 , 4 is a zero of R , and since
x โ 3 is a factor of R x ( )= x^2 โ 7 x + 12 , 3 is a zero of R.
If P ( x ) is a polynomial of degree n > 0, such that
P x ( ) = a x ( โ c 1 )( x โ c 2 ) ... ( x โ cn )(with a โ 0)
then the numbers c 1 , c 2 ,... c (^) n are the zeros of P. These zeros need not all be different. If the factor x โ c appears k times in the complete factorization of P ( x ), then we say that c is a zero of multiplicity k.
The following theorem and its consequences will be used to help us graph polynomials.
Intermediate Value Theorem for Polynomials:
If P is a polynomial function and P(a) and P(b) have opposite signs, then there exists at least one value c between a and b for which P(c) = 0.
The figure below graphically demonstrates this theorem.
One important consequence of this theorem is that between any two successive zeros, the values of a polynomial are either all positive or all negative. That is, between two successive zeros the graph of a polynomial lies entirely above or entirely below the x - axis.
So, to sketch the graph of P , we first find all the zeros of P. Then we choose test points between (and to the right and left of) successive zeros to determine whether P(x) is positive or negative on each interval determined by the zeros.
Guidelines for Graphing Polynomial Functions:
- Zeros: Factor the polynomial to find all its real zeros; these are the x - intercepts of the graph.
- Test Points: Make a table of values for the polynomial. Include test points to determine whether the graph of the polynomial lies above or below the x -axis on the intervals determined by the zeros. Include the y -intercept in the table
- End Behavior: Determine the end behavior of the polynomial.
- Graph: Plot the intercepts and other points you found in the table. Sketch a smooth curve that passes through these points and exhibits the required end behavior.
Example 4 (Continued):
Solution:
Step 4: Finally, we plot the points from the table and connect the points by a smooth curve to complete the graph.
Example 5: Sketch the graph of the function Q x ( ) = x^4^ โ 4 x^2. Make sure your graph
shows all intercepts and exhibits the proper end behavior.
Solution:
Step 1: In order to find all the real zeros of Q x ( ), we must first factor
it completely.
2 2 2
Q x x x x x x x x
Factor^2 Difference of Squares
x
Step 2: Since Q x ( ) = x^2^ ( x + 2 )( x โ 2 )the zeros are , and
. Thus, the x -intercepts are
x = 0 x = โ 2 x = 2 x = 0 , x = โ 2 and x = 2.
Example 5 (Continued):
Solution:
Step 3: Now we will make a table of values of Q x ( ), making sure we
choose test points between (and to the left and right of)
successive zeros, and include the y -intercept, Q ( 0 )= 0.
Step 4: Next we determine the end behavior. Since Q is of even degree (degree 4) and its leading coefficient is positive, it has the following end behavior:
y โ โ as x โ โ and y โ โ as x โ โโ
Step 5: Finally, we plot the points from the table and connect the points by a smooth curve to complete the graph.
Example 6 (Continued):
Solution:
Step 2: Now we will make a table of values of R ( x ), making sure we
choose test points between (and to the left and right of)
successive zeros, and the y -intercept, R ( 0 )= โ 8.
Step 3: Since R is of odd degree (degree 5) and its leading coefficient is negative, it has the following end behavior:
y โ โโ as x โ โ and y โ โ as x โ โโ
Step 4: Finally, we plot the points from the table and connect the points by a smooth curve to complete the graph.
