Bell-ringer

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Bell-ringer

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

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7.1 Introduction to Polynomials
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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
  monomials variables.
• i.e. x3y2z is of degree 6 because x3y2z1 3 2
  1 6

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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.

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Examples v. Non-examples

• Examples
• 5x 4
• x4 3x3 2x2 5x -1
• v7x2 3x 5

• Non examples
• x3/2 2x 1
• 3/x2 4x3 3x 13
• 3vx x4 3x3 9x 7

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Classification

• We classify polynomials by
• the number of terms or monomials it
  contains
• AND
• by its degree.

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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

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Classification of a Polynomial
Degree Name Example






n 0
constant
3
n 1
linear
5x 4
n 2
quadratic
2x2 3x - 2
cubic
n 3
5x3 3x2 x 9
quartic
3x4 2x3 8x2 6x 5
n 4
-2x5 3x4 x3 3x2 2x 6
n 5
quintic
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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.

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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.

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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.

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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

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Example
Add (-3x4y3 6x3y3 6x2 5xy5 1) (5x5
3x3y3 5xy5)
-3x4y3 6x3y3 6x2 5xy5 1
5x5 - 3x3y3 - 5xy5
1
5x5
3x4y3
3x3y3
6x2
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Example
Subtract. (2x2y2 3xy3 4y4) - (x2y2 5xy3
3y 2y4)
2x2y2 3xy3 4y4
- x2y2 5xy3 3y 2y4
x2y2
8xy3
2y4
3y
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Evaluating Polynomials

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

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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.

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Exploring Graphs of Polynomial Functions Activity

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

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Graphs of Polynomial Functions
Graph each function below.
Function Degree of U-turns in the graph





2
1
y x2 x - 2
y 3x3 12x 4
3
2
y -2x3 4x2 x - 2
3
2
y x4 5x3 5x2 x - 6
4
3
y x4 2x3 5x2 6x
4
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Make a conjecture about the degree of a function
and the of U-turns in the graph.
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Graphs of Polynomial Functions
Graph each function below.
Function Degree of U-turns in the graph



3
0
y x3
y x3 3x2 3x - 1
3
0
y x4
4
1
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.
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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.

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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.

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Homework

• Page 429-430 12-48 by 3s.