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
PowerPoint Presentation: Examples Find the degree, leading coefficient and constant terms of: Polynomial Degree Leading Coefficient Constant Term
PowerPoint Presentation: 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
PowerPoint Presentation: 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.
PowerPoint Presentation: Division of Polynomials – – – – Divide: Answer:
PowerPoint Presentation: Another Division of Polynomials Divide: – + – + + – + – +
PowerPoint Presentation: And So… Answer:
PowerPoint Presentation: You Try… Divide: Answer: Divide: Answer:
PowerPoint Presentation: 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.
PowerPoint Presentation: 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:
PowerPoint Presentation: An Example: Use the remainder theorem to calculate f ( 3 ) for the polynomial function . We divide and get… …and therefore…
PowerPoint Presentation: 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 :
PowerPoint Presentation: An Example: Show that and are factors of the polynomial. Thus by the Factor Theorem are factors of f ( x ) : and and
PowerPoint Presentation: 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.
PowerPoint Presentation: 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…
PowerPoint Presentation: From a Graph: The picture shows a third degree polynomial function, f ( x ). Find the equation of f ( x ).
PowerPoint Presentation: 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