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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