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Chapter 11: Polynomials
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Friday, February 27, 2009
Section 11-1Introduction to polynomials
Friday, February 27, 2009
Degree of a polynomial:
Terms of the polynomial:
Friday, February 27, 2009
Degree of a polynomial:
Terms of the polynomial:
The largest exponent of the variable
Friday, February 27, 2009
Degree of a polynomial:
Terms of the polynomial:
The largest exponent of the variable
Each collection of a variable and coefficient in the polynomial; separated by + or -
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Polynomial in x:
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Polynomial in x:
An expression, where n is a positive integer and an ≠ 0
Friday, February 27, 2009
Polynomial in x:
An expression, where n is a positive integer and an ≠ 0
anxn + an− 1xn− 1 + ...+ a2x
2 + a1x + a0
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Standard form:
Leading coefficient
Friday, February 27, 2009
Standard form:
Leading coefficient
A polynomial that is written in descending order of degree
Friday, February 27, 2009
Standard form:
Leading coefficient
A polynomial that is written in descending order of degree
The number that is with the variable of the highest degree; an
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Example 1a. Expand and write in standard form:
Friday, February 27, 2009
Example 1a. Expand and write in standard form:
(5x3 − 6)2
Friday, February 27, 2009
Example 1a. Expand and write in standard form:
(5x3 − 6)2
= 25x6 − 60x 3 + 36
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Example 1a. Expand and write in standard form:
b. What is the degree?
(5x3 − 6)2
= 25x6 − 60x 3 + 36
Friday, February 27, 2009
Example 1a. Expand and write in standard form:
b. What is the degree?
6
(5x3 − 6)2
= 25x6 − 60x 3 + 36
Friday, February 27, 2009
Example 1a. Expand and write in standard form:
b. What is the degree?
6
c. What is the leading coefficient?
(5x3 − 6)2
= 25x6 − 60x 3 + 36
Friday, February 27, 2009
Example 1a. Expand and write in standard form:
b. What is the degree?
6
c. What is the leading coefficient?
25
(5x3 − 6)2
= 25x6 − 60x 3 + 36
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Some special types of polynomials
Degree 1:
Degree 2:
Degree 3:
Degree 4:
Degree 5+:
Friday, February 27, 2009
Some special types of polynomials
Degree 1:
Degree 2:
Degree 3:
Degree 4:
Degree 5+:
Linear
Friday, February 27, 2009
Some special types of polynomials
Degree 1:
Degree 2:
Degree 3:
Degree 4:
Degree 5+:
Linear
Quadratic
Friday, February 27, 2009
Some special types of polynomials
Degree 1:
Degree 2:
Degree 3:
Degree 4:
Degree 5+:
Linear
Quadratic
Cubic
Friday, February 27, 2009
Some special types of polynomials
Degree 1:
Degree 2:
Degree 3:
Degree 4:
Degree 5+:
Linear
Quadratic
Cubic
Quartic
Friday, February 27, 2009
Some special types of polynomials
Degree 1:
Degree 2:
Degree 3:
Degree 4:
Degree 5+:
Linear
Quadratic
Cubic
Quartic
Fifth degree, sixth degree, etc.
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Polynomial Function:
Friday, February 27, 2009
Polynomial Function:
A function P(x) where P(x) is a polynomial
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Example 2
a. Find P(-1)
P(x) = x 5 − 4x 4 + x 2 − 5x + 50
Friday, February 27, 2009
Example 2
a. Find P(-1)
P(x) = x 5 − 4x 4 + x 2 − 5x + 50
P(−1) = (−1)5 − 4(−1)4 + (−1)2 − 5(−1) + 50
Friday, February 27, 2009
Example 2
a. Find P(-1)
P(x) = x 5 − 4x 4 + x 2 − 5x + 50
P(−1) = (−1)5 − 4(−1)4 + (−1)2 − 5(−1) + 50
= −1− 4 + 1+ 5 + 50
Friday, February 27, 2009
Example 2
a. Find P(-1)
P(x) = x 5 − 4x 4 + x 2 − 5x + 50
P(−1) = (−1)5 − 4(−1)4 + (−1)2 − 5(−1) + 50
= −1− 4 + 1+ 5 + 50 = 51
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b. Graph P(x) in your graphing calculator. Set your window to -5 ≤ x ≤ 5 and -60 ≤ y ≤ 60.
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b. Graph P(x) in your graphing calculator. Set your window to -5 ≤ x ≤ 5 and -60 ≤ y ≤ 60.
Friday, February 27, 2009
b. Graph P(x) in your graphing calculator. Set your window to -5 ≤ x ≤ 5 and -60 ≤ y ≤ 60.
Friday, February 27, 2009
b. Graph P(x) in your graphing calculator. Set your window to -5 ≤ x ≤ 5 and -60 ≤ y ≤ 60.
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Example 3On Matt Mitarnowski’s eighteenth birthday, he inherited $5000 that he invested in a savings plan at 8% annual yield. He then set out a plan to save for a house, saving an extra $2000 at the end of each year. He needs a down payment of $20000. Will he have enough after 6 years?
Friday, February 27, 2009
Example 3On Matt Mitarnowski’s eighteenth birthday, he inherited $5000 that he invested in a savings plan at 8% annual yield. He then set out a plan to save for a house, saving an extra $2000 at the end of each year. He needs a down payment of $20000. Will he have enough after 6 years?
This is just a multi-part compound interest problem.
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Matt gets $5000 that he invests annually at 8% interest. He invests this for 6 years.
Friday, February 27, 2009
Matt gets $5000 that he invests annually at 8% interest. He invests this for 6 years.
5000(1.08)6
Friday, February 27, 2009
Matt gets $5000 that he invests annually at 8% interest. He invests this for 6 years.
5000(1.08)6
Then, for each of the next five years, he invests an additional $2000.
Friday, February 27, 2009
Matt gets $5000 that he invests annually at 8% interest. He invests this for 6 years.
5000(1.08)6
Then, for each of the next five years, he invests an additional $2000.
+ 2000(1.08)5
Friday, February 27, 2009
Matt gets $5000 that he invests annually at 8% interest. He invests this for 6 years.
5000(1.08)6
Then, for each of the next five years, he invests an additional $2000.
+ 2000(1.08)5 + 2000(1.08)4
Friday, February 27, 2009
Matt gets $5000 that he invests annually at 8% interest. He invests this for 6 years.
5000(1.08)6
Then, for each of the next five years, he invests an additional $2000.
+ 2000(1.08)5 + 2000(1.08)4 + 2000(1.08)3
Friday, February 27, 2009
Matt gets $5000 that he invests annually at 8% interest. He invests this for 6 years.
5000(1.08)6
Then, for each of the next five years, he invests an additional $2000.
+ 2000(1.08)5 + 2000(1.08)4
+ 2000(1.08)2
+ 2000(1.08)3
Friday, February 27, 2009
Matt gets $5000 that he invests annually at 8% interest. He invests this for 6 years.
5000(1.08)6
Then, for each of the next five years, he invests an additional $2000.
+ 2000(1.08)5 + 2000(1.08)4
+ 2000(1.08)2
+ 2000(1.08)3
+ 2000(1.08)
Friday, February 27, 2009
Matt gets $5000 that he invests annually at 8% interest. He invests this for 6 years.
5000(1.08)6
Then, for each of the next five years, he invests an additional $2000.
+ 2000(1.08)5 + 2000(1.08)4
+ 2000(1.08)2
+ 2000(1.08)3
+ 2000(1.08) + 2000
Friday, February 27, 2009
Matt gets $5000 that he invests annually at 8% interest. He invests this for 6 years.
5000(1.08)6
Then, for each of the next five years, he invests an additional $2000.
+ 2000(1.08)5 + 2000(1.08)4
+ 2000(1.08)2
+ 2000(1.08)3
+ 2000(1.08) + 2000
= $7934.37 + $2938.65 + $2720.97 + $2519.42 + $2332.80 + $2160 + $2000
Friday, February 27, 2009
Matt gets $5000 that he invests annually at 8% interest. He invests this for 6 years.
5000(1.08)6
Then, for each of the next five years, he invests an additional $2000.
+ 2000(1.08)5 + 2000(1.08)4
+ 2000(1.08)2
+ 2000(1.08)3
+ 2000(1.08) + 2000
= $7934.37 + $2938.65 + $2720.97 + $2519.42 + $2332.80 + $2160 + $2000
= $22,606.21
Friday, February 27, 2009
Homework
Friday, February 27, 2009
Homework
p. 677 #1-23, skip 18
“Don’t go around saying the world owes you a living. The world owes you nothing. It was here first.” - Mark Twain
Friday, February 27, 2009
