Transcript 6.1 Polynomial Functions

6.1 Polynomial Functions
At the end of this lesson, you should
be able to:
 classify polynomials
 perform basic operations with polynomials
 use polynomial operations to solve real-life
problems
Vocabulary
is an expression that is either
A monomial __________________________
a_________________________________
real number, a variable, or the
__________________________________
product of a real number and a variable
monomial or the sum of
monomials___________________
A polynomial
DEFINITION OF POLYNOMIAL IN X
Polynomial functions are functions
that can be written in this form:
f(x) = anxn + an-1xn-1 + ... + a1x + a0
NOTEWORTHY:
The value of n must be a nonnegative integer.
The coefficients, as they are called, are an, an-1, ...,
a1, a0. These are real numbers.
The degree of the polynomial function is the highest
value for n where an is not equal to 0.
The terms in the polynomial are shown in descending
order by degree. This order illustrates the standard
form of a polynomial.
Polynomials with one, two, or three terms are called,
monomials, binomials, and trinomials, respectively.
Standard form of a polynomial:
P(x) =
Leading
Coefficient
2 x3 -5x2 - 2x + 5
Cubic
Term
Quad.
Term
Linear
Term
Constant
Term
CLASSIFYING POLYNOMIALS
Polynomials can be classified by
degree
number of terms
_________
and by ______________.
A polynomial of more then three terms
has no special name.
Complete the chart below:
Degree
Name
Using
Degree
Polynomial
Example
0
6
1
2
x+5
2x2
3
4
5
2x3 – 5x2 + 3x
x4 + 5
-2x5 + 6x3 –x + 1
Quartic
Quintic
Number Name
Of
by
Terms
Terms
ADDING, SUBTRACTING, AND
MULTIPLYING POLYNOMIALS
Note: We will be
using horizontal
format.
To add or subtract
polynomials, we
combine like terms
(equivalent
variables) using
either horizontal
or vertical format.
Adding and Subtracting
Polynomials
Examples:
(y2 - 3y + 6) + (y - 3y 2 + y 3)
(5x2 + 2x +1) - ( 3x2 – 4x –2)
Review: Special Product
Patterns
Sum and Difference
(u + v) (u - v) =
Example:
(2x + 3) (2x – 3) =
u2 – v2
Review: Special Product
Patterns
Square of a Binomial Example
(u + v)2 = u2 + 2uv + v2
(u - v)2 =
u2 - 2uv + v2
♥ Perfect Cubes:
Note: Being familiar with perfect cubes
will make quick mental math out of cubing
a binomial!
13 = 1
__
73 =343
__
23 =8__ 33 = 27
__ 43 = 64
__
83 =512
__ 93 = 729
__ 103 = _
53 =125
__
203 = _
63 =216
__
303 = _
Learn-by-♥ stuff!
Special Product Patterns
Cube of a Binomial
u3 + 3u2v + 3uv2 + v3
(u + v)3 =
Example:
(x + 5)3 =
(u - v)3 =
Example:
(x - 3)3 =
u3 - 3u2v + 3uv2 - v3
CHECKING FOR UNDERSTANDING:
Write each polynomial in standard form, then
classify it by degree and number of terms.
(x + 3) (x – 7)
9x 6 y 5 - 7x 4 y3 + 3x 3 y 4 + 17x – 4
(2x + 5)3
(2c – 3) (2c + 4) (c + 1)
(2x + 5y) + (3x – 2y)
(3x 3 + 3x2 – 4x + 5) + (x 3 – 2x2 + x – 4)
A POLYNOMIAL MODEL FOR
VOLUME
2x + 1
x+2
A rectangular box has sides
whose lengths (in inches) are
(2x + 1), (x + 2), and
(x – 2). Write a polynomial, in
standard form, for the
volume of the box. Then find
x-2
the volume of the box when x
is 5 inches.
MODELING DATA WITH A
POLYNOMIAL FUNCTION
x
0
5
10
y
10.1
2.8
8.1
15
20
16.0 17.8
Determine whether a linear, quadratic, or cubic
model best fits the data, by using the LinReg,
QuadReg, and CubicReg options of your graphing
calculator to find the best-fitting model for each
polynomial classifications.
Final Checks for Understanding
1.
Perform the indicated operations,
then classify the resulting polynomial
by degree and number of terms:
(3x
3
+ 3x2 – 4x + 5) + (x 3 – 2x2 + x – 4)
2. Find the area of the blue region:
x
2x + 1
x
4x
HOMEWORK
POLYNOMIAL FUNCIONS WS, PLUS
TEXT PAGES ___________________