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Bell-ringer. Holt Algebra II text page 431 #72-75, 77-80. 7.1 Introduction to Polynomials. Definitions. Monomial - is an expression that is a number, a variable, or a product of a number and variables. i.e. 2, y, 3x, 45x 2 … Constant - is a monomial containing no variables.

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Bell ringer

Bell-ringer

  • Holt Algebra II text page 431 #72-75, 77-80


7 1 introduction to polynomials

7.1 Introduction to Polynomials


Definitions

Definitions

  • Monomial - is an expression that is a number, a variable, or a product of a number and variables.

    • i.e. 2, y, 3x, 45x2…

  • Constant - is a monomial containing no variables.

    • i.e. 3, ½, 9 …

  • Coefficient - is a numerical factor of a monomial.

    • i.e. 3x, 12y, 2/3x3, 7x4 …

  • Degree - is the sum of the exponents of a monomial’s variables.

    • i.e. x3y2z is of degree 6 because x3y2z1 = 3 + 2+ 1 = 6


Definitions1

Definitions

  • Polynomial- is a monomial or a sum of terms that are monomials.

    • These monomials have variables which are raised to whole-number exponents.

    • The degree of a polynomial is the same as that of its term with the greatest degree.


Examples v non examples

Examples v. Non-examples

  • Examples

    5x + 4

    x4 + 3x3 – 2x2 + 5x -1

    √7x2 – 3x + 5

  • Non – examples

    x3/2 + 2x – 1

    3/x2 – 4x3 + 3x – 13

    3√x +x4 +3x3 +9x +7


Classification

Classification

  • We classify polynomials by…

    …the number of terms or monomials it contains

    AND

    … by its degree.


Classification of polynomials

Classification of Polynomials

  • Classifying polynomials by the number of terms…

    monomial: one term

    binomial: two terms

    trinomial: three terms

    Poylnomial: anything with four or more terms


Bell ringer

Classification of a Polynomial

n = 0

constant

3

linear

n = 1

5x + 4

quadratic

n = 2

2x2 + 3x - 2

cubic

n = 3

5x3 + 3x2 – x + 9

quartic

3x4 – 2x3 + 8x2 – 6x + 5

n = 4

n = 5

-2x5 + 3x4 – x3 + 3x2 – 2x + 6

quintic


Compare the two expressions

Compare the Two Expressions

  • How do these expressions compare to one another?

    3(x2 -1) - x2 + 5x and 5x – 3 + 2x2

  • How would it be easier to compare?

  • Standard form - put the terms in descending order by degree.


Examples

Examples

  • Write each polynomial in standard form, classifying by degree and number of terms.

    1). 3x2 – 4 + 8x4

    = 8x4+ 3x2 – 4

    quartic trinomial

    2). 3x2 +2x6 - + x3 - 4x4 – 1 –x3

    = 2x6- 4x4 + 3x2 – 1

    6th degree polynomial with four terms.


Adding subtracting polynomials

Adding & Subtracting Polynomials

  • To add/subtract polynomials, combine like terms, and then write in standard form.

  • Recall: In order to have like terms, the variable and exponent must be the same for each term you are trying to add or subtract.


Examples1

Examples

  • Add the polynomial and write answer in standard form.

    1). (3x2 + 7 + x) - (14x3 + 2 + x2 - x) =

    =- 14x3 + (3x2 - x2) +(x -x) + (7- 2)

    = - 14x3 + 2x2 + 5


Example

Example

Add

(-3x4y3 + 6x3y3 – 6x2 + 5xy5 + 1) + (5x5 – 3x3y3 – 5xy5)

-3x4y3 + 6x3y3 – 6x2 + 5xy5 + 1

5x5 - 3x3y3 - 5xy5

+ 1

5x5

– 3x4y3

+ 3x3y3

– 6x2


Example1

Example

Subtract.

(2x2y2 + 3xy3 – 4y4) - (x2y2 – 5xy3 + 3y – 2y4)

= 2x2y2 + 3xy3 – 4y4

- x2y2 + 5xy3 – 3y + 2y4

= x2y2

+ 8xy3

– 2y4

– 3y


Evaluating polynomials

Evaluating Polynomials

  • Evaluating polynomials is just like evaluating any function.

    *Substitute the given value for each variable and then do the arithmetic.


Application

Application

  • The cost of manufacturing a certain product can be approximated by f(x) = 3x3 – 18x + 45, where x is the number of units of the product in hundreds. Evaluate f(0) and f(200) and describe what they represent.

    • f(0) = 45 represents the initial cost before manufacturing any products f(200) = 23,996,445 represents the cost of manufacturing 20,000 units of the product.


Exploring graphs of polynomial functions activity

Exploring Graphs of Polynomial Functions Activity

  • Copy the table on page 427

  • Answer/complete each question/step.


Graphs of polynomial functions

Graphs of Polynomial Functions

Graph each function below.

2

1

y = x2 + x - 2

3

2

y = 3x3 – 12x + 4

3

2

y = -2x3 + 4x2 + x - 2

4

3

y = x4 + 5x3 + 5x2 – x - 6

4

3

y = x4 + 2x3 – 5x2 – 6x

Make a conjecture about the degree of a function and the # of “U-turns” in the graph.


Graphs of polynomial functions1

Graphs of Polynomial Functions

Graph each function below.

3

0

y = x3

3

0

y = x3 – 3x2 + 3x - 1

4

1

y = x4

Now make another conjecture about the degree of a function and the # of “U-turns” in the graph.

The number of “U-turns” in a graph is less than or equal to one less than the degree of a polynomial.


Now you

Now You

  • Graph each function. Describe its general shape.

    • P(x) = -3x3 – 2x2 +2x – 1

    • An S-shaped graph that always rises on the left and falls on the right.

    • Q(x) = 2x4 – 3x2 – x + 2

    • W-shape that always rises on the right and the left.


Check your understanding

Check Your Understanding

  • Create a polynomial.

  • Trade polynomials with the second person to your left.

  • Put your new polynomial in standard form then…

    …identify by degree and number of terms

    …identify the number of U - turns.

  • Turn the papers in with both names.


Homework

Homework

  • Page 429-430 #12-48 by 3’s.


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