Polynomial Functions

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An introduction to Polynomial Functions and Polynomial Division for Precalculus I at Elmira College

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Section 4.2: 

Section 4.2 Polynomial Functions Dr. Joseph Kolacinski Assistant Professor Elmira College

Polynomials in One Variable: 

A polynomial is an algebraic expression of the form: Polynomials in One Variable where n > 0 is an integer and each a k is a real number called the coefficient of x k . Examples:

Some Terminology: 

A polynomial is said to be in standard form if the terms are written in decreasing order. Some Terminology The degree of a polynomial is the largest power of x that appears with a non-zero coefficient. The leading coefficient , is the coefficient of the highest degree term. is the constant term . Example: degree: 4 coefficients: 2, 0, −3, 1, −5 Leading Coefficient: 2 Constant Term: −5

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Examples Find the degree, leading coefficient and constant terms of: Polynomial Degree Leading Coefficient Constant Term

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More Terminology A function defined by a polynomial is said to be a polynomial function. We’ve already encountered some particular polynomial functions. Degree 0: constant functions Degree 1: linear functions Degree 2: quadratic functions

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Arithmetic of Polynomials In order to work effectively with polynomials, you need to be proficient at doing arithmetic with them. Since this is covered in detail in College Algebra, we won’t re-do all that work. If you’re having difficulties you can review addition, subtraction, multiplication and factoring of polynomials in Section 0.4 of our text. Come see me if you need help. We will talk about division of polynomials which looks like long division of whole numbers.

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Division of Polynomials – – – – Divide: Answer:

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Another Division of Polynomials Divide: – + – + + – + – +

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And So… Answer:

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You Try… Divide: Answer: Divide: Answer:

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The Division Algorithm If a polynomial f ( x ) is divided by a non-zero polynomial d ( x ), then there is a quotient polynomial q ( x ) and a remainder polynomial r ( x ) such that Earlier we calculated We can write this as Notice that d ( x ) is a factor of f ( x ) exactly when r ( x ) = 0.

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The Remainder Theorem If f ( x ) is a polynomial, then f ( c ) is the remainder of the division . For example, consider: Calculate f (3): Divide f ( x ) by x – c and use the remainder:

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An Example: Use the remainder theorem to calculate f ( 3 ) for the polynomial function . We divide and get… …and therefore…

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The Factor Theorem The number c is a zero of a polynomial f ( x ) if and only if x - c is a factor of our polynomial. For example, consider: 2 is a zero of f : x - c is a factor of f :

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An Example: Show that and are factors of the polynomial. Thus by the Factor Theorem are factors of f ( x ) : and and

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Suppose that we know Help With Factoring: Notice that, with our current methods we are unable to factor And the new piece is something we can factor, so… So… is a factor, which means we can divide it out.

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We know , and are factors. Another Example: Find the polynomial function p ( x ), that is degree three, has zeroes at 1, -3 and 4 and that has Since , we can solve for a and we get So…

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From a Graph: The picture shows a third degree polynomial function, f ( x ). Find the equation of f ( x ).

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You Try… Find the equation of the fourth degree polynomial function q ( x ) shown in the picture.

Homework:: 

Homework: For Tuesday, 29 November Section 4.1, all assigned problems Section 4.2, assigned problems up to #35