Download File aa u1 day 01 student notes polynomial functions add subtract

AA U1 (3.1): Add and Subtract Polynomial Functions
AAPR 1 Understand that polynomials form a system analogous to the integers, namely they are closed under the
operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials
Essential Question: How do you add and subtract polynomials?
REVIEW:
The round smiley faces are a ___________. No matter what
operation is performed on round smiley faces, another round
smiley face will be created. Thus, there are always only round
smiley faces in the box.
A set is closed (under an operation) if and only if the operation
on two elements of the set produces another element of the
set. If an element outside the set is produced, then the
operation is not closed.
Ex: If you multiply two real numbers, you will get another real
number.
Ex. If you add two even numbers (from the set of even
numbers), is the sum even?
Since the sum (the answer) is always _________, the set of
even numbers is ___________ under the operation of addition.
Does this mean even numbers are closed for all operations?
When you find even ________ that does not work, the set is
____________ under that operation. The even numbers are
not closed under division.
How about multiplication and subtraction?
You Try: Add ( 4x
3
+ 11x 2 + 8x + 5) + (7x 2 - 5x + 9).
REFLECT
1a. Do you get the same results whether you add polynomials
vertically or horizontally? Why or why not?
1b. Is the sum of two polynomials always another polynomial?
Explain.
1c. Is the sum of two polynomials of degree 5 always a
polynomial of degree 5? Give an example to explain your answer.
You Try: Subtract (4 + 7x
2
) - (-5x 2 - 3x + 2).
REFLECT
2a. How is subtracting polynomials similar to subtracting
integers?
2b. In part A, you leave a gap in the polynomial 9x2 + 2 when you
write the subtraction problem vertically. Why?
2c. Is the difference of two polynomials always another
polynomial? Explain.
REFLECT
3a. Is it possible to solve this problem without adding the
polynomials? Explain.
3b. Explain how you can use the given information to estimate
how many more male high school students than female high
school students there were in the United States in 2007.
You Try: For the presidential elections from 1980 to 2008, the
votes cast for the Democratic candidate can be modeled by
D(x) = 0.00230x3 – 0.0625x2 + 1.17x + 34.9 where x is the
number of years since 1980 and D(x) is the number of
Democratic votes cast in millions. The votes cast for the
Republican candidate in these elections can be modeled by
R(x) = –0.00140x4 + 0.0809x3 – 1.41x2 + 7.29x + 43.5 where x is
the number of years since 1980 and R(x) is the number of
Republican votes cast in millions. Write a model for the total
Democratic and Republican votes cast in the presidential
elections from 1980 to 2008, and use it to estimate the total
Democratic and Republican votes cast in the 2000 election.
REMINDER FROM LAST YEAR
Classification:
DEGREE NAME
GRAPH
0
# OF
TERMS
1
1
2
2
3
3
4 or more
NAME
4
5
6 or
more
Polynomial?
1.
2.
3.
3 x + 2x – x – 7
5x + 2x3 – 2x2 -5
x5–4x3–x5+3x2+4x3
4.
3
5

6
2
x
x
5.
x5 x3

5
3
Number
of terms
Degree
Leading
Coefficient
constant
Ex 4. Let f(x) = 10x3 + 9x2 – 19x + 6
a) State the degree.
b) Classify by the number of terms
c) State the leading coefficient
d) Evaluate f(x) for x = -3
Ex 5. Let g(x) = 3x3y2
a) State the degree.
b) Classify by the number of terms
c) State the leading coefficient
d) Evaluate g(x) for x = 1
You Try: Let h(x) = –10x2 + 2
a) State the degree.
b) Classify by the number of terms
c) State the leading coefficient
d) Evaluate h(x) for x = -1