Polynomial Functions

Frank RagusaAman GrewalNiall Dennehy

Definition of Polynomial Functions

Example:

P(x) = anxn + an-1xn-1 + … + a1x1 +a0

An ≠ 0

Review

The zeros of a function are also the solutions.

A number is said to be a zero when P(x) = 0. The degree of a polynomial is determined by

the largest degree in that function:– 5th degree polynomial

A point on the graph that separates an increasing portion from a decreasing portion is called a turning point.

Polynomial functions are generally written in descending order.

Properties of Polynomial Graphs(Theorem 1)

A polynomial graph has a domain of all real numbers.

It has no sharp corners. It has at most n real zeros (n is the

degree of the polynomial). Has at most n-1 turning points. Increases or decreases without bound

as x ∞ and as x -∞.

Example of a Polynomial Graph

This can be a 4th degree polynomial function.– It has 3 turning points– It has 4 (n) zeros– It’s domain is all real

numbers– It has no sharp

corners– It increases without

bound

Absolute Value Graph

This is not a polynomial graph because it has a sharp corner.

Left and Right Behavior of Polynomial Functions

(Theorem 2)

When the degree of the polynomial is even, odd, positive, or negative the graph of P(x) increases without bound as x ∞ and as x -∞.

Division Algorithm(Theorem 3)

For each polynomial P(x) of a degree greater than 0 and each number r, there exists a unique polynomial Q(x) of 1 degree less than P(x) and a unique number r that is the remainder.

One can use synthetic division in order to find the remainder.

One simple example is 7 divided by 2 is 3 remainder 1.

The Remainder Theorem

The remainder theorem states that the remainder r of P(x) divided by x is P(r).

In order to find the remainder, you can use synthetic division or plug that number into the polynomial function.

(Refer to example 1).

The Factor Theorem

If 0 is the remainder, then that divisor is a factor of the dividend.

If r is a zero of the polynomial P(x), then x-r is a factor of P(x). Conversely, if x-r is a factor of P(x), then r is a zero of P(x).

One simple example is 6 divided by 2 is 3 remainder 0.

(Refer to example 2).

Fundamental Theorem

If P(x) is degree n, then P(x) has n zeros.

For example, a 5th degree polynomial has 5 zeros that can be real, imaginary, or both.

Least Upper Bound

The least upper bound is the first positive integer where the solutions to synthetic division are all positive.

0 counts as either positive or negative. In the least upper bound, 0 would be positive.

(Refer to example 3).

Greatest Lower Bound

The first negative integer where the solution to synthetic division alternates between positive and negative.

0 counts as either positive or negative. (Refer to example 4).

Location Theorem

On an interval (a,b), if P(a) is positive and P(b) is negative (or vice versa) then a zero of P(x) is on the interval (a,b).

An example would be if P(2) is negative and P(3) is positive, then there is a zero between 2 and 3. However, 2 and 3 are not zeros.

Linear Factors Theorem

Every polynomial of a degree greater than 0 can be factored as a product of n linear factors.

Multiplicity is the number of times a factor is repeated.

Linear and Quadratic Factors Theorem

If P(x) is a polynomial with a degree greater than 0, then P(x) can be factored as a product of linear factors (real zeros) and quadratic factors (with real and imaginary zeros).

Zeros of Even or Odd Multiplicity Theorem

When a zero is also a turning point, it has an even multiplicity (e.g. multiplicity 2).

When a zero is not a turning point, it has an odd multiplicity (e.g. multiplicity 1).

Real Zeros and Polynomials of Odd Degree Theorem

Every polynomial of odd degree with real coefficients has at least one real zero.

Finding Real and Imaginary Zeros

Graph the polynomial function. Once graphed, hit 2nd, trace, zero in order to

find the zeros. Factor out as many real zeros as possible in

order to get the function to a quadratic. Once the function is a quadratic, use the

quadratic formula or complete the square in order to find the imaginary zeros.

(Refer to example 6).

Real World Example: Profits

Business analysts often use polynomial functions in order to predict the profit of a certain company.

The example on the following slide is one that predicts profit.

Profit Prediction

A manager estimates that if the company charges p dollars for their new product, where 0 ≤ p ≤ 200, then the revenue from the product will be r(p) = 2,000p – 10p^2 dollars each week. According to this model, for which of the following values of p would the company's weekly revenue for the product be the greatest?

A.     10 B.   20 C.    50 D.   100 E.   200

Profit Prediction: Solution

To solve this problem, you can graph it. On the y= menu on your calculators, enter the equation. You must reformat your window in order to see the entire graph. Once your window is reformatted, find the maximum value of the parabola and you will find that the 100 (D) is the maximum value.

Both 0 and 200 are the x intercepts of the parabola, and thus have the least value.

Works Cited Page

Landsberger, Joseph. "Parabola." Polynomial Functions. 12 December 2009. Google Images. <http://people.richland.edu/james/lecture/m116/functions/cubic.gif>.

Green, Michael. "Absolute Value." Functions. 12 December 2009. Google Images. <http://hotmath.com/images/gt/lessons/genericalg1/abs_value_graph.gif>.

Wheeling, James. "Functions." Polynomial Functions. 12 December 2009. Google Images. <http://www.biology.arizona.edu/biomath/tutorials/polynomial/graphics/PolynomialNegOdd.gif>.

Johnson, William. "Polynomials." Functions. 12 December 2009. Google Images. <http://tutorial.math.lamar.edu/Classes/Alg/GraphingPolynomials_files/image008.gif>.