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Math 71A

5.1 – Introduction to Polynomials and Polynomial Functions

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A number or a number multiplied by variables is called a ________________(also called a ________).

Polynomials

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A number or a number multiplied by variables is called a ________________(also called a ________).

Polynomials

monomial

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A number or a number multiplied by variables is called a ________________(also called a ________).

Polynomials

monomialterm

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Ex 1.Which of the following are monomials? 17

Polynomials

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Polynomials

Ex 1.Which of the following are monomials? 17

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The sum of the exponents of the variables is called the ________________ of a monomial.

The numerical factor in a monomial is called the ___________________.

Polynomials

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The sum of the exponents of the variables is called the ________________ of a monomial.

The numerical factor in a monomial is called the ___________________.

Polynomials

degree

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The sum of the exponents of the variables is called the ________________ of a monomial.

The numerical factor in a monomial is called the ___________________.

Polynomials

degree

coefficient

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Ex 2.What are the degrees and coefficients of the following monomials?

PolynomialsMonomial Degree Coefficient

17

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Ex 2.What are the degrees and coefficients of the following monomials?

PolynomialsMonomial Degree Coefficient

2 -4

17

12

Ex 2.What are the degrees and coefficients of the following monomials?

PolynomialsMonomial Degree Coefficient

2 -4

13 5

17

13

Ex 2.What are the degrees and coefficients of the following monomials?

PolynomialsMonomial Degree Coefficient

2 -4

13 5

4 2

17

14

Ex 2.What are the degrees and coefficients of the following monomials?

PolynomialsMonomial Degree Coefficient

2 -4

13 5

4 2

1

17

15

Ex 2.What are the degrees and coefficients of the following monomials?

PolynomialsMonomial Degree Coefficient

2 -4

13 5

4 2

1

2 1

17

16

Ex 2.What are the degrees and coefficients of the following monomials?

PolynomialsMonomial Degree Coefficient

2 -4

13 5

4 2

1

2 1

1 -1

17

17

Ex 2.What are the degrees and coefficients of the following monomials?

PolynomialsMonomial Degree Coefficient

2 -4

13 5

4 2

1

2 1

1 -1

17 0 17

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One or more monomials added together is called a ___________________.

ex:

The highest degree of all terms in a polynomial is called the _________________ of the polynomial.

ex: What is degree of above polynomial?

Polynomials

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One or more monomials added together is called a ___________________.

ex:

The highest degree of all terms in a polynomial is called the _________________ of the polynomial.

ex: What is degree of above polynomial?

Polynomials

polynomial

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One or more monomials added together is called a ___________________.

ex:

The highest degree of all terms in a polynomial is called the _________________ of the polynomial.

ex: What is degree of above polynomial?

Polynomials

polynomial

degree

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One or more monomials added together is called a ___________________.

ex:

The highest degree of all terms in a polynomial is called the _________________ of the polynomial.

ex: What is degree of above polynomial?

Polynomials

polynomial

degree

2+3=𝟓

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A polynomial with 2 terms is called a ____________________.

A polynomial with 3 terms is called a ____________________.

Polynomials

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A polynomial with 2 terms is called a ____________________.

A polynomial with 3 terms is called a ____________________.

Polynomials

binomial

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A polynomial with 2 terms is called a ____________________.

A polynomial with 3 terms is called a ____________________.

Polynomials

binomial

trinomial

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Ex 3.Add:

Adding Polynomials

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Ex 3.Add:

Adding Polynomials

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Ex 3.Add:

Adding Polynomials

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Ex 4.Subtract:

Subtracting Polynomials

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Ex 4.Subtract:

Subtracting Polynomials

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Ex 4.Subtract:

Subtracting Polynomials

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Here’s a polynomial function: And here’s what looks like:

Polynomial Functions

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In general, polynomial functions are __________ (i.e.

only have rounded curves with no sharp corners) and

________________ (i.e. don’t have breaks and can be

drawn without lifting pencil).

Polynomial Functions

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In general, polynomial functions are __________ (i.e.

only have rounded curves with no sharp corners) and

________________ (i.e. don’t have breaks and can be

drawn without lifting pencil).

Polynomial Functions

smooth

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In general, polynomial functions are __________ (i.e.

only have rounded curves with no sharp corners) and

________________ (i.e. don’t have breaks and can be

drawn without lifting pencil).

Polynomial Functions

smooth

continuous

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Also, polynomial functions always either rise or fall as you go to the left or right (except for constant functions).

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This is called the end behavior of a function, and depends on the ______________ (the term with _________ degree).

Ex 5.Find the leading term and leading coefficient of the following polynomial functions. Leading term: ______ Leading coefficient: ______ Leading term: ______ Leading coefficient: ______

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This is called the end behavior of a function, and depends on the ______________ (the term with _________ degree).

Ex 5.Find the leading term and leading coefficient of the following polynomial functions. Leading term: ______ Leading coefficient: ______ Leading term: ______ Leading coefficient: ______

leading term

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This is called the end behavior of a function, and depends on the ______________ (the term with _________ degree).

Ex 5.Find the leading term and leading coefficient of the following polynomial functions. Leading term: ______ Leading coefficient: ______ Leading term: ______ Leading coefficient: ______

leading termhighest

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This is called the end behavior of a function, and depends on the ______________ (the term with _________ degree).

Ex 5.Find the leading term and leading coefficient of the following polynomial functions. Leading term: ______ Leading coefficient: ______ Leading term: ______ Leading coefficient: ______

leading termhighest

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This is called the end behavior of a function, and depends on the ______________ (the term with _________ degree).

Ex 5.Find the leading term and leading coefficient of the following polynomial functions. Leading term: ______ Leading coefficient: ______ Leading term: ______ Leading coefficient: ______

leading termhighest

𝟕 𝒙𝟔

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This is called the end behavior of a function, and depends on the ______________ (the term with _________ degree).

Ex 5.Find the leading term and leading coefficient of the following polynomial functions. Leading term: ______ Leading coefficient: ______ Leading term: ______ Leading coefficient: ______

leading termhighest

𝟕 𝒙𝟔 𝟕

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This is called the end behavior of a function, and depends on the ______________ (the term with _________ degree).

Ex 5.Find the leading term and leading coefficient of the following polynomial functions. Leading term: ______ Leading coefficient: ______ Leading term: ______ Leading coefficient: ______

leading termhighest

𝟕 𝒙𝟔 𝟕

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This is called the end behavior of a function, and depends on the ______________ (the term with _________ degree).

Ex 5.Find the leading term and leading coefficient of the following polynomial functions. Leading term: ______ Leading coefficient: ______ Leading term: ______ Leading coefficient: ______

leading termhighest

𝟕 𝒙𝟔 𝟕

−𝟑𝒙𝟒

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This is called the end behavior of a function, and depends on the ______________ (the term with _________ degree).

Ex 5.Find the leading term and leading coefficient of the following polynomial functions. Leading term: ______ Leading coefficient: ______ Leading term: ______ Leading coefficient: ______

leading termhighest

𝟕 𝒙𝟔 𝟕

−𝟑𝒙𝟒 −𝟑