Roller coaster polynomials 

Polynomials Sec 9.1.1

Learning Targets  Vocabulary  Operations between polynomials  Introduction to graphs of polynomials

Definitions  Polynomial comes from poly- (meaning "many") and -nomial (in this case meaning "term")... so it means “many terms”  Term: A number, a variable, or the product/quotient of numbers/variables.

Polynomial

A Term has 3 Components: Coefficient: can be any real number… including zero. Variable Exponent: Can only be positive integers: 0,1,2, 3, These components are very important!!!

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Naming a Polynomial  We can classify a polynomial based on how many terms it has: Polynomial 7 5x + 2 4x 2 + 3x - 4 6x # Terms # Terms Name monomial binomial trinomial binomial

Naming Cont.  Quadrinomial (4 term) and quintinomial (5 term) also exist, but those names are not often used.  Polynomials Can Have Lots and Lots of Terms  Polynomials can have as many terms as needed, but not an infinite number of terms. infinite For more than 3 terms say: “a polynomial with n terms” or “an n- term polynomial” 11x 8 + x 5 + x 4 - 3x 3 + 5x “a polynomial with 6 terms” – or – “a 6- term polynomial”

Degree of a Term The degree of a term is determined by the exponent of the variable. Term 3 4x -5x 2 18x 5 Degree of Term

Naming a Polynomial  We can also classify a polynomial based on its highest degree: Polynomial 7 5x + 2 4x 2 + 3x - 4 6x Degree # Degree Name Constant Linear Quadratic Cubic

Putting it All Together Name cubic monomial quadratic monomial constant monomial linear binomial cubic trinomial quadratic trinomial 4 th degree binomial Polynomial -14x x 2 7x - 2 3x 3 + 2x - 8 2x 2 - 4x + 8 x 4 + 3

Standard Form of a Polynomial A polynomial written so that the degree of the terms decreases from left to right and no terms have the same degree.

Not Standard 6x + 3x x - x+ 5x 4 x x 1 + x 2 + x + x 3 Standard 3x 2 + 6x - 2 5x 4 - 4x x + 10 x 3 + x 2 + x + 1

Finding the Leading Terms Degree

Fill in the following table: Leading CoefficientDegreeEnd Behaviors Positive Negative Positive Negative Degree Even Odd End Behavior

Leading CoefficientDegreeEnd Behaviors Positive Negative Positive Negative End BehaviorDegree Even Odd

Types of Roots  Polynomial solutions are made up of complex roots  A root is where the polynomial’s graph will intersect with the x-axis  A complex root describes two different types of roots:  Real Roots  Imaginary Roots (we will get to these next week)

Root Classifications  We classify the type of Real Root based on the degrees of each term and how it interacts with the x-axis.  Types:  Single Root  Double Root  Triple Root  And so on…

Examples:  Single Roots

Examples:  Double Roots

Examples:  Triple Roots

You Try  Classify each type of root:

Desmos  More on degree and type of roots 

Practice

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Turns In a Graph  What determines the number of turns the graph of a polynomial will have?  End Behavior  Degree of the Leading Term  Degrees of each factor, or the types of roots  The maximum number of turns a polynomial can have is (n-1) where n is the degree of the leading term

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