Roller coaster polynomials

https://www.youtube.com/watch?v=fmZ8jDCVIwc

PolynomialsSec 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 Example of Polynomial

Terms

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!!!

NOT ALLOWED Negative exponents:

Variables in the denominator:

Check In Which of the following is a polynomial:

1) 5

Naming a Polynomial We can classify a polynomial based on how

many terms it has:Polynomial

75x + 2

4x2 + 3x - 46x3 - 18

# Terms

1232

# Terms Name

monomialbinomialtrinomialbinomial

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.

For more than 3 terms say:“a polynomial with n terms” or

“an n-term polynomial”

11x8 + x5 + x4 - 3x3 + 5x2 - 3“a polynomial with 6 terms” – or – “a

6-term polynomial”

Degree of a TermThe degree of a term is determined by

the exponent of the variable.

Term

3

4x

-5x2

18x5

Degree of Term

0

1

2

5

Naming a Polynomial We can also classify a polynomial based on

its highest degree:Polynomial

75x + 2

4x2 + 3x - 46x3 - 18

Degree

0123

# Degree Name

ConstantLinear

QuadraticCubic

Putting it All TogetherName

cubic monomialquadratic monomialconstant monomial

linear binomialcubic trinomial

quadratic trinomial4th degree binomial

Polynomial-14x3

-1.2x2

-17x - 2

3x3+ 2x - 82x2 - 4x + 8

x4 + 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 + 3x2 - 2

15 - 3x - x+ 5x4

x + 10 + x

1 + x2 + x + x3

Standard

3x2 + 6x - 2

5x4 - 4x + 15

2x + 10

x3 + x2 + x + 1

Finding the Leading Terms Degree In order to find the highest degree we can

add up all of the factors exponents.

This sum will always result in the highest degree of the leading term

Using this information we can determine how a graph will behave as x approaches and

Fill in the following table:

Leading Coefficient Degree End BehaviorsPositiveNegativePositiveNegative

DegreeEvenEvenOddOdd

End Behavior

End Behavior

Leading Coefficient Degree End BehaviorsPositiveNegativePositiveNegative

End BehaviorDegreeEvenEvenOddOdd

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

f(x)=(x-2)(x+2)

-4 -3 -2 -1 1 2 3 4

x

y

Examples: Double Roots

f(x)=.05((x-2)^2)((x+2)^2)

-4 -3 -2 -1 1 2 3 4

x

y

Examples: Triple Roots

f(x)=.05((x-2)^3)((x+2)^3)

-4 -3 -2 -1 1 2 3 4

x

y

You Try Classify each type of root:

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9x

y

Desmos More on degree and type of roots https://

www.desmos.com/calculator/zj7vtcpmpw

Practice Sketch the following polynomials, describe the

end behavior and classify the roots:

1)

2)

3)

#1

-19-18-17-16-15-14-13-12-11-10-9-8-7-6-5-4-3-2-1 1 2 3 4 5 6 7 8 9101112131415161718192021

x

y

This is only a sketch

#2

-19-18-17-16-15-14-13-12-11-10-9-8-7-6-5-4-3-2-1 1 2 3 4 5 6 7 8 9101112131415161718192021

x

y

This is only a sketch

#3

-18-17-16-15-14-13-12-11-10-9-8-7-6-5-4-3-2-1 1 2 3 4 5 6 7 8 9101112131415161718192021

x

y

This is only a sketch

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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