Transcript 7.1 Introduction to Polynomials

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, 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 wholenumber exponents.
 The degree of a polynomial is the same as that of
its term with the greatest degree.
Examples v. Non-examples
 Examples
 Non – examples
5x + 4
x3/2 + 2x – 1
x4 + 3x3 – 2x2 + 5x -1
3/x2 – 4x3 + 3x – 13
√7x2 – 3x + 5
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
Degree
Name
Example
n=0
constant
3
n=1
linear
5x + 4
n=2
quadratic
2x2 + 3x - 2
n=3
cubic
5x3 + 3x2 – x + 9
n=4
quartic
3x4 – 2x3 + 8x2 – 6x + 5
n=5
quintic
-2x5 + 3x4 – x3 + 3x2 – 2x + 6
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)
5x5
-3x4y3 + 6x3y3 – 6x2 + 5xy5 + 1
- 3x3y3
- 5xy5
5x5 – 3x4y3 + 3x3y3 – 6x2
+1
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.
Function
Degree # of U-turns
in the graph
y = x2 + x - 2
2
1
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
3
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.
Function
Degree # of U-turns
in the graph
y = x3
3
0
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.
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.