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Bell-ringer

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

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

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
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

- We classify polynomials by…
…the number of terms or monomials it contains

AND

… by its degree.

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

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

- 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

- 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

- 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.

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

Add

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

-3x4y3 + 6x3y3 – 6x2 + 5xy5 + 1

5x5 - 3x3y3 - 5xy5

+ 1

5x5

– 3x4y3

+ 3x3y3

– 6x2

Example

Subtract.

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

= 2x2y2 + 3xy3 – 4y4

- x2y2 + 5xy3 – 3y + 2y4

= x2y2

+ 8xy3

– 2y4

– 3y

Evaluating Polynomials

- Evaluating polynomials is just like evaluating any function.
*Substitute the given value for each variable and then do the arithmetic.

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

- Copy the table on page 427
- Answer/complete each question/step.

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 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

- 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

- 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

- Page 429-430 #12-48 by 3’s.

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