Unit 2: Polynomials
Notes from 9/22/14
Naming Polynomials and Operations on Polynomials
Naming Polynomials and Operations on Polynomials
Poly: means many
Nomial: means terms --> therefore, polynomial means "many terms"!
Naming a Polynomial
Always done in TWO PARTS! : The highest degree AND the number of terms
Examples: 3x^2 + 4x + 3 is a quadratic trinomial
4x - 2 is a linear binomial
-10 is a constant monomial
x^7 + 4x^3 + 2x -1 is a 7th degree polynomial with 4 terms
For chart on naming click here.
Nomial: means terms --> therefore, polynomial means "many terms"!
Naming a Polynomial
Always done in TWO PARTS! : The highest degree AND the number of terms
Examples: 3x^2 + 4x + 3 is a quadratic trinomial
4x - 2 is a linear binomial
-10 is a constant monomial
x^7 + 4x^3 + 2x -1 is a 7th degree polynomial with 4 terms
For chart on naming click here.
Notes from 9/24/14
Binomial Theorem
Binomial Theorem
Binomial Theorem: an elegant and efficient way to expand binomials to a power
Click here for my notes from today. Please remember that when you square (4x), you must distribute the square. When you square 4x, you get 256x^2 NOT 4x^2.
Check your Pascal's Triangle with the one below to be sure you have done it correctly!
Click here for my notes from today. Please remember that when you square (4x), you must distribute the square. When you square 4x, you get 256x^2 NOT 4x^2.
Check your Pascal's Triangle with the one below to be sure you have done it correctly!
Notes from 9/29
Long Division of Polynomials
Long Division of Polynomials
Click here for my powerpoint slides.
|
Notes from 10/1
Sums and Differences of Cubes
Sums and Differences of Cubes
Click here for my notes on Differences and Sums of Cubes
Notes from 10/6
Factoring and Solving Polynomials
Click here for my slides on solving polynomials from 10/6/14
Notes from 10/8
Solving Polynomials and Quadratic Form
Solving Polynomials and Quadratic Form
Vocabulary
Prime Polynomial: A polynomial that cannot be factored (think of a prime number which is a number that cannot be divided by anything else, therefore it has no FACTORS)
Quadratic Form: An expression that can be rewritten as au^2+bu+c for any numbers a, b, and c, where u is some expression in x.
Click here for my slides from 10/8/14
Prime Polynomial: A polynomial that cannot be factored (think of a prime number which is a number that cannot be divided by anything else, therefore it has no FACTORS)
Quadratic Form: An expression that can be rewritten as au^2+bu+c for any numbers a, b, and c, where u is some expression in x.
- au^2+bu+c is called the QUADRATIC FORM of the original expression
- Remember this will only work when it is a TRINOMIAL when the degree of the first term is twice the degree of the second term
- Example: 6x^6+x^3+2
Click here for my slides from 10/8/14
Notes from 10/13
Graphing Polynomials
Graphing Polynomials
End behavior: What happens to the graph as x approaches positive and negative infinity
Odd polynomial: A polynomial with an odd degree
Odd polynomial: A polynomial with an odd degree
- As x --> negative infinity, f(x) --> negative infinity AND as x --> positive infinity, f(x) --> positive infinity OR
- As x --> negative infinity, f(x) --> positive infinity AND as x--> positive infinity, f(x) --> negative infinity
- As x --> negative infinity, f(x) --> positive infinity AND as x --> positive infinity, f(x) --> positive infinity OR
- As x --> negative infinity, f(x) --> negative infinity AND as x --> positive infinity, f(x) --> negative infinity
Notes from 10/15
Fundamental Theorem of Algebra
Fundamental Theorem of Algebra
Fundamental Theorem of Algebra: every polynomial equation with degree greater than zero has at least one root in the set of complex numbers
- Root: the x-intercept of a polynomial graph; Example: 3 and -2 are the roots of (x-3)(x+2)
- Set of complex numbers: ALL numbers REAL and IMAGINARY
- *****The fundamental theorem tell us: A polynomial equation of degree n has EXACTLY n roots in the set of complex numbers, including repeated roots; Example x^5 is a polynomial of degree 5 and therefore has EXACTLY 5 roots (in this case, all 5 roots are 0)