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Polynomial Functions Unit 4

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Polynomial Functions

Unit 4

Polynomials

• Monomial—a number, variable, or product of numbers and variables all raised to whole number powers

• Polynomial Expression--a monomial or sum of monomials.

• Polynomial Function—function that is defined by a polynomial expression

• Leading coefficient—coefficient of highest powered term

• Degree of polynomial—power of highest powered variable

Page 3: Polynomial Functions Unit 4. Polynomials Monomial—a number, variable, or product of numbers and variables all raised to whole number powers Polynomial

Standard Form

The standard form of a polynomial function arranges the terms by degree in descending numerical order

A polynomial function P(x) in standard form is:

Where n is a nonnegative integer and are real numbers. Ex:

Page 4: Polynomial Functions Unit 4. Polynomials Monomial—a number, variable, or product of numbers and variables all raised to whole number powers Polynomial

Classifying Polynomials by degree• d=0 Constant

• d=1 Linear

• d=2 Quadratic

• d=3 Cubic

• d=4 Quartic

• d=5 Quintic xxx

xxxx

2289

234

Page 5: Polynomial Functions Unit 4. Polynomials Monomial—a number, variable, or product of numbers and variables all raised to whole number powers Polynomial

Classifying by Number of Terms

• Monomial—one term

• Binomial—two terms

• Trinomial—three terms

• N-nomial—n terms

Page 6: Polynomial Functions Unit 4. Polynomials Monomial—a number, variable, or product of numbers and variables all raised to whole number powers Polynomial

Classify by degree and number of terms.

Polynomial Function

• Polynomial equation used to represent a function

3210)(

254)(23

xxxxP

xxxf

Page 8: Polynomial Functions Unit 4. Polynomials Monomial—a number, variable, or product of numbers and variables all raised to whole number powers Polynomial

Graphs of Polynomial Functions

• Constant Linear Quadratic• Show graphs with positive and negative LC

Page 9: Polynomial Functions Unit 4. Polynomials Monomial—a number, variable, or product of numbers and variables all raised to whole number powers Polynomial

End Behavior of Graphs

As As

As x gets bigger or smaller, what happens to the function value?

Page 10: Polynomial Functions Unit 4. Polynomials Monomial—a number, variable, or product of numbers and variables all raised to whole number powers Polynomial

Graphs of Polynomial Functions

• Quadratic: D:________R: ________Zeros: _______Inc: _______Dec: _______as as

Page 11: Polynomial Functions Unit 4. Polynomials Monomial—a number, variable, or product of numbers and variables all raised to whole number powers Polynomial

Graphs of Polynomial Functions

• Cubic: D:________R: ________Zeros: _______Inc: _______Dec: _______as as

Page 12: Polynomial Functions Unit 4. Polynomials Monomial—a number, variable, or product of numbers and variables all raised to whole number powers Polynomial

Graphs of Polynomial Functions

• Quartic: D:________R: ________Zeros: _______Inc: _______Dec: _______as as

Page 13: Polynomial Functions Unit 4. Polynomials Monomial—a number, variable, or product of numbers and variables all raised to whole number powers Polynomial

Graphs of Polynomial Functions

• Quintic: D:________R: ________Zeros: _______Inc: _______Dec: _______as as

Page 14: Polynomial Functions Unit 4. Polynomials Monomial—a number, variable, or product of numbers and variables all raised to whole number powers Polynomial

End Behavior of Graphs+Lead Coefficient -Lead Coefficient

Even Degree

Odd Degree

Page 15: Polynomial Functions Unit 4. Polynomials Monomial—a number, variable, or product of numbers and variables all raised to whole number powers Polynomial

End Behavior of Graphs

As As

Page 16: Polynomial Functions Unit 4. Polynomials Monomial—a number, variable, or product of numbers and variables all raised to whole number powers Polynomial

End Behavior of Graphs

As As

Page 17: Polynomial Functions Unit 4. Polynomials Monomial—a number, variable, or product of numbers and variables all raised to whole number powers Polynomial

To sketch the graph

• Determine the end behavior.

• Determine the x intercepts (where y=0)

020

)2(0

2)(2

xandx

xxxf

Page 18: Polynomial Functions Unit 4. Polynomials Monomial—a number, variable, or product of numbers and variables all raised to whole number powers Polynomial

Turning Points

A polynomial function of degree n has at most n-1 turning points.

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