Factoring Review:
Factoring is a tool used in mathematics to break functions down into their various components so when they are multiplied together they result in the original expression or function. In order to be successful in this course, you will need to recall how to factor various bionomials and trinomials. Here's a handy list of the types of factoring you will use regularly:
Characteristics of a Function?
1) General Overview: http://www.purplemath.com/modules/fcns.htm
2) Terminology
a) End Behaviour - What happens to the y-values as x approaches positive or negative infinity
b) Symmetry - even = line symmetry about the y-axis
odd = point symmetry about the origin (can be rotated 180 degrees to conicide with itself)
c) Quadrants - Functions extend from one quadrant to another (read left to right)
d) Domain/Range - set of all x/y values
e) Interval Notation - 3 methods = number line, inequality, brackets "(" up to and not including, "[" including
Characteristics of Polynomial Functions:
- variable
- term(s)
- exponent on the variable
- terms connected by +/-
- can be graphed
- can be put in decreasing orders of x
- can be coefficients and/or constant terms
- exponents are positive and whole numbers
- the degree is the highest exponent in the polynomial function. For example, in the function f(x) = 3x^4+ x^3 - 2, the degree is 4. - the leading coefficient is the number before the x. For example, in the function f(x) = 3x^4+ x^3 - 2, the leading coefficient is 3. Things We Know
1) A negative leading coefficient means the function ends in quadrant 4 and vice versa.
2) If degree of function is odd, the function starts and ends at opposite sides of the x-axis.
Refer to: http://www.hfcrd.ab.ca/cyberhigh/Math/Math20P/Unit03PF/PolynRev.pdf for greater detail
Examples
1) Which are polynomial functions? Why or why not? g(x) = sinx * is not a polynomial functionf(x) = 2x^4k(x) = x^3 - 5x^2 + 6x -8m(x) = 3^x *is not a polynomial function because x is the exponent
2) Come up with examples of polynomial functions whose graph extends from:
a) Quadrant 3 to 1: y = 3x^3
b) Quadrant 2 to 4: y = 3(-x)^3
c) Quadrant 3 to 4: y = -3x^2
d) Quadrant 2 to 1: y = 4x^2
September 17: Finite Differences and Polynomial Functions!
updated by: Shagufta Panchbhaya
Finite Differences:
For a polynomial function of degree n, where n is a positive integer (ex: f(x) = x^n) , then nth differences:
1) are equal
2) have the same sign as the leading coefficient
3) the value of the nth differences is equal to: (leadingcoefficient)(n!)
Factorials:
- indicated by a !
- n! means the product of all the numbers from n to 1
Ex: 5! = 5 x 4 x 3 x 2 x 1 = 120
Ex: 26! = 26 x 25 x 24 x 23 x 22 ...... x 1
Example:
Find the degree of the polynomial, the sign, and value of hte leading coefficient.
x
y
FIRST
SECOND
THIRD
-3
-36
-2
-12
24
-1
-2
10
-14
0
0
2
-8
6
1
0
0
-2
6
2
4
4
4
6
3
18
14
10
6
4
48
40
26
6
degree --> 3 (the third finite differences are all the same, or constant: 6)
sign --> + (the third finite differences are positive)
leading coefficient --> 1
To calculate the leading coefficient:
(leadingcoefficient)(n!) = constant value
(LC)(3!) = 6
LC = 6/3!
LC = 6/6
LC = 1
When calculating finite differences remember: number under - number above
References:
Handout : What is the relationship between finite differences and the equation of a polynomial function?
Handout: Problem (using Excel and regression)
For more information about calculating finite differences: http://www.math-mate.com/chapter37.shtml
Homework:
Handout: Key Features of Graphs of Polynomials
Handout: 1.7.2: Factoring in our Graphs
We can use the factored form of a polynomial function to graph it.
The key information to use is:
1. Where are the x- intercepts?
ex. f(x)= x(x-2)(x=3) x=0, 2, -3
2.What does the leading coefficient tell us? If it is positive the functions ends in quadrant 1 (the functions ends up). If it is negative the functions ends in quadrant 4 (the function ends down).
3. What exponent is on each factor? If the exponent is 1 the functions passes through the x-intercept, if it is 2 the function touches then "bounces off" the x-axis and if it is 3 the function acts like a cubic ( "spends more time").
ex. f(x)=(x-1)(x+4)2(x-6)3
4. Even/ Odd Check. If the functions has an even degree it will start and end on the same side of the x-axis. If the function has an odd degree it will start and end on opposite sides of the x-axis.
Homework: Handouts: "What's my Polynomial Name?", "Factoring Polynomials- Finding Zeroes of Polynomials", and "Unit #1:Polynomial Functions".
Terms from "1.7.3: What's My Polynomial Name?" homework sheet:
- inflection point = turning point
- multiplicity = degree of factor
Finding the leading coefficient given one point
Given point (2,2) on a curve, facing down, going through -2, bouncing off 1, and through 3.
f(x) = -a (x+2) (x-1)^2 (x-3)
Sub in the x and y value for (2,2):
2 = -a (2+2) (2-1)^2 (2-3)
= -a (4) (1) (-1)
= 4a
a = 1/2
cautionary tale: a might not always be the value of 1 *check for the leading coefficient if given a point.
Even and odd revisited
1) Even functions : f(x) = f(-x)
Ex: Is g(x) even?
g(x) = x^2 - 8
g(-x) = (-x)^2 - 8
= x^2 - 8
Therefore, yes, g(x) is an even function.
2) Odd functions : f(x) = - f(-x)
Ex: If f(x) odd? f(x) = x^3 - 4x - f(-x) = -[(-x)^3 - 4(-x)] = -(-x^3 + 4x) = x^3 - 4x Therefore, yes, f(x) is an odd function.
*Can a function be both even and odd? Ex) y=0 This is a rare case, though, and once determined even or odd, it's not necessary to disprove the opposite.
*Can a function be neither? Yes, if it does not have even or odd symmetry, it will be neither.
Homework: p 3 # 1-3, 13, 15 p 8 # 1-6, 9 p 12 #1-5, 7, 8 p 16 #4, 5, 8
September 22: Quest!
updated by: Patrick Lasagna
Not much to say is there? We had a Quest today, it covered everything we learned up until this point.
Friday, September 24, 2010: Dividing Polynomial Functions
Updated by: Eva Klimova
Sorry OSCSS Math students, my word document would not copy and paste properly onto this page, so here is the document link. Hope it helps! I pasted two links just in case one does not work.
Solving polynomial equations = finding their factors! (x intercepts)
Ex) Solve!
a) x^3 - x^2 - 2x = 0
x (x^2 - x - 2) = 0
x (x - 2)(x + 1) = 0 x = 0, 2 and -1
b) 3x^3 + x^2 - 12x - 4 = 0
factors of 4 = + 4, + 2, + 1
f (2) = 3(2)^3 + (2)^2 - 12(2) - 4
f (2) = 0 -----> therefore (x-2) is a factor!
2 | 3 1 -12 -4 6+ 14+ 4+
3 7 2 0
3x^2 + 7x + 2
3x^2 + 6x + x + 2
3x (x +2) + 1 (x + 2) f(x) = (3x + 1)(x + 2)(x - 2) x = -1/3 and + 2 What if our polynomial is NOT factorable????
See exercises relating to silos and whales :)
Friday, October 1st, 2010 : Solving Polynomial Inequalities
By: Eva Klimova
Homework: Solving Polynomial Inequalities Assignment DUE Tuesday, October 5, 2010. Tuesday October 5 2010
updated by Sarah Drury
Today we had our Polynomial Functions unit test!
The Teamwork Tuesday task for this week is: Conjugate Roots (Teams of 3) - In class to be presented on chart paper
Two roots of a quartic equation are complex conjugates of each other. What can this quartic equation be? What are its roots? Look up any words you don't understand :) http://www.analyzemath.com/complex/complex_numbers.html
updated by: Gillian Evans
Factoring Review:
Factoring is a tool used in mathematics to break functions down into their various components so when they are multiplied together they result in the original expression or function. In order to be successful in this course, you will need to recall how to factor various bionomials and trinomials. Here's a handy list of the types of factoring you will use regularly:
Characteristics of a Function?
1) General Overview: http://www.purplemath.com/modules/fcns.htm
2) Terminology
a) End Behaviour - What happens to the y-values as x approaches positive or negative infinity
b) Symmetry - even = line symmetry about the y-axis
odd = point symmetry about the origin (can be rotated 180 degrees to conicide with itself)
c) Quadrants - Functions extend from one quadrant to another (read left to right)
d) Domain/Range - set of all x/y values
e) Interval Notation - 3 methods = number line, inequality, brackets "(" up to and not including, "[" including
Homework:
Complete both of the following:
Bring this one to class to hand-in:
September 15: Polynomial Functions!
updated by: Shagufta Panchbhaya
Characteristics of Polynomial Functions:
- variable
- term(s)
- exponent on the variable
- terms connected by +/-
- can be graphed
- can be put in decreasing orders of x
- can be coefficients and/or constant terms
- exponents are positive and whole numbers
Examples of Polynomial Functions:
f(x) = x^2 + 2x -1
f(x) = 3x^4+ x^3 - 2
f(x) = x^5+ x^4 + x^3+ x^2+ x+ 1
f(x) = -x^7 + 2x^2
f(x) = x^2 + 2x -1
f(x) = 1/2 x^2 + 3/2x -1/5
f(x) = 7
- the degree is the highest exponent in the polynomial function. For example, in the function
f(x) = 3x^4+ x^3 - 2, the degree is 4.
- the leading coefficient is the number before the x. For example, in the function f(x) = 3x^4+ x^3 - 2, the leading coefficient is 3.
Things We Know
1) A negative leading coefficient means the function ends in quadrant 4 and vice versa.
2) If degree of function is odd, the function starts and ends at opposite sides of the x-axis.
Refer to: http://www.hfcrd.ab.ca/cyberhigh/Math/Math20P/Unit03PF/PolynRev.pdf for greater detail
Examples
1) Which are polynomial functions? Why or why not?
g(x) = sinx * is not a polynomial functionf(x) = 2x^4k(x) = x^3 - 5x^2 + 6x -8m(x) = 3^x *is not a polynomial function because x is the exponent
2) Come up with examples of polynomial functions whose graph extends from:
a) Quadrant 3 to 1: y = 3x^3
b) Quadrant 2 to 4: y = 3(-x)^3
c) Quadrant 3 to 4: y = -3x^2
d) Quadrant 2 to 1: y = 4x^2
References
- handout Power Functions and End Behaviour of Power Functions
- http://www.purplemath.com/modules/polyends.htm
Homework:
Complete pg 4 # 5-7, 14 in textbook
Refer to my scanned notes if needed: MathNote.jpg
September 17: Finite Differences and Polynomial Functions!
updated by: Shagufta Panchbhaya
Finite Differences:
For a polynomial function of degree n, where n is a positive integer (ex: f(x) = x^n) , then nth differences:
1) are equal
2) have the same sign as the leading coefficient
3) the value of the nth differences is equal to: (leadingcoefficient)(n!)
Factorials:
- indicated by a !
- n! means the product of all the numbers from n to 1
Ex: 5! = 5 x 4 x 3 x 2 x 1 = 120
Ex: 26! = 26 x 25 x 24 x 23 x 22 ...... x 1
Example:
Find the degree of the polynomial, the sign, and value of hte leading coefficient.
sign --> + (the third finite differences are positive)
leading coefficient --> 1
To calculate the leading coefficient:
(leadingcoefficient)(n!) = constant value
(LC)(3!) = 6
LC = 6/3!
LC = 6/6
LC = 1
When calculating finite differences remember: number under - number above
References:
Handout : What is the relationship between finite differences and the equation of a polynomial function?
Handout: Problem (using Excel and regression)
For more information about calculating finite differences: http://www.math-mate.com/chapter37.shtml
Homework:
Handout: Key Features of Graphs of Polynomials
Handout: 1.7.2: Factoring in our Graphs
Refer to my scanned notes if needed: MathNote!.jpg
September 20: Graphing Polynomial Functions
Updated by Wade WalkerWe can use the factored form of a polynomial function to graph it.
The key information to use is:
1. Where are the x- intercepts?
ex. f(x)= x(x-2)(x=3) x=0, 2, -3
2.What does the leading coefficient tell us? If it is positive the functions ends in quadrant 1 (the functions ends up). If it is negative the functions ends in quadrant 4 (the function ends down).
3. What exponent is on each factor? If the exponent is 1 the functions passes through the x-intercept, if it is 2 the function touches then "bounces off" the x-axis and if it is 3 the function acts like a cubic ( "spends more time").
ex.
f(x)=(x-1)(x+4)2(x-6)3
4. Even/ Odd Check. If the functions has an even degree it will start and end on the same side of the x-axis. If the function has an odd degree it will start and end on opposite sides of the x-axis.
Homework: Handouts: "What's my Polynomial Name?", "Factoring Polynomials- Finding Zeroes of Polynomials", and "Unit #1:Polynomial Functions".
Additional references:http://www.youtube.com/watch?v=PmBNhKRhpqE&feature=related
http://zonalandeducation.com/mmts/functionInstitute/polynomialFunctions/graphs/polynomialFunctionGraphs.html
September 21: Even and odd revisited
Updated by Joanne KimTerms from "1.7.3: What's My Polynomial Name?" homework sheet:
- inflection point = turning point
- multiplicity = degree of factor
Finding the leading coefficient given one point
Given point (2,2) on a curve, facing down, going through -2, bouncing off 1, and through 3.
f(x) = -a (x+2) (x-1)^2 (x-3)
Sub in the x and y value for (2,2):
2 = -a (2+2) (2-1)^2 (2-3)
= -a (4) (1) (-1)
= 4a
a = 1/2
cautionary tale: a might not always be the value of 1 *check for the leading coefficient if given a point.
Even and odd revisited
1) Even functions : f(x) = f(-x)
Ex: Is g(x) even?
g(x) = x^2 - 8
g(-x) = (-x)^2 - 8
= x^2 - 8
Therefore, yes, g(x) is an even function.
2) Odd functions : f(x) = - f(-x)
Ex: If f(x) odd?
f(x) = x^3 - 4x
- f(-x) = -[(-x)^3 - 4(-x)]
= -(-x^3 + 4x)
= x^3 - 4x
Therefore, yes, f(x) is an odd function.
More examples:
f(x) = 2x^4 - 5x^2 + 4 (Even)
f(x) = -3x^5 + 9x^3 + 2x (Odd)
f(x) = -x^4 + x^2
f(x) = -cos x
f(x) = x^3
f(x) = -3x^5 + x^3
*Can a function be both even and odd?
Ex) y=0
This is a rare case, though, and once determined even or odd, it's not necessary to disprove the opposite.
*Can a function be neither?
Yes, if it does not have even or odd symmetry, it will be neither.
For more information, visit:
http://www.purplemath.com/modules/fcnnot3.htm
For additional practice, visit:
http://www.analyzemath.com/function/even_odd.html
Homework:
p 3 # 1-3, 13, 15
p 8 # 1-6, 9
p 12 #1-5, 7, 8
p 16 #4, 5, 8
September 22: Quest!
updated by: Patrick Lasagna
Not much to say is there? We had a Quest today, it covered everything we learned up until this point.
Friday, September 24, 2010: Dividing Polynomial Functions
Updated by: Eva Klimova
Sorry OSCSS Math students, my word document would not copy and paste properly onto this page, so here is the document link. Hope it helps! I pasted two links just in case one does not work.
Monday, September 27, 2010:The Factor Theorem
Updated by: Clinton D'Silva
So similar to Eva I can't copy and paste my word document so heres the link!!
Have fun
HOMEWORK
pg 35 # 2-4, 9-12
Overview of THE FACTOR THEOREM
http://www.purplemath.com/modules/factrthm.htm
Need Help with Synthetic Division?
http://www.purplemath.com/modules/synthdiv4.htm
Tuesday, September 28, 2010: Factor Difference + sum of cubes
Updated by: Joanne Kim
Factor: x^3 - y^3, where x is a variable, and y is a random number standing for something.
Check f(y) : (y)^3 - y^3 = 0 therefore, (x-y) is a factor.
Dividing out (x-y)
y| 1 0 0 -y^3
y y^2 y^3
1 y y^2 0
therefore, x^2 + xy + y^2 remains.
Factoring the difference of cubes: x^3 - y^3 = (x-y) (x^2 + xy + y^2)
Ex) x^3 - 125 = x^3 - 5^3 = (x-5) (x^2+5x+25)
Factor x^3 + y^3
Check f(-y) : (-y)^3 + y^3 = 0 therefore, (x+y) is a factor.
Dividing out (x+y)
-y | 1 0 0 y^3
-y y^2 -y^3
1 -y y^2 0
therefore, x^2 - xy + y^2 remains.
Factoring the sum of cubes: x^3 + y^3 = (x+y) (x^2 - xy + y^2)
Ex) x^3 + 64 = x^3 + 4^3 = (m+4) (m^2 - 4m + 16)
Teamwork Tuesdays!
Q: Find a polynomial that passes through points (-2, -1) (-1, 7) (2, -5) and (3, -1)
Homework: Polynomial review, Factoring sum and difference of cubes.
Extra information:
http://www.purplemath.com/modules/specfact2.htm
Wednesday September 29 2010
Solving Polynomials Equations!!
by:Federico Palacios
Solving polynomial equations = finding their factors! (x intercepts)
Ex) Solve!
a) x^3 - x^2 - 2x = 0
x (x^2 - x - 2) = 0
x (x - 2)(x + 1) = 0
x = 0, 2 and -1
b) 3x^3 + x^2 - 12x - 4 = 0
factors of 4 = + 4, + 2, + 1
f (2) = 3(2)^3 + (2)^2 - 12(2) - 4
f (2) = 0 -----> therefore (x-2) is a factor!
2 | 3 1 -12 -4
6+ 14+ 4+
3 7 2 0
3x^2 + 7x + 2
3x^2 + 6x + x + 2
3x (x +2) + 1 (x + 2)
f(x) = (3x + 1)(x + 2)(x - 2)
x = -1/3 and + 2
What if our polynomial is NOT factorable????
See exercises relating to silos and whales :)
Friday, October 1st, 2010 : Solving Polynomial Inequalities
By: Eva Klimova
Homework: Solving Polynomial Inequalities Assignment DUE Tuesday, October 5, 2010.
Tuesday October 5 2010
updated by Sarah Drury
Today we had our Polynomial Functions unit test!
The Teamwork Tuesday task for this week is:
Conjugate Roots (Teams of 3) - In class to be presented on chart paper
Two roots of a quartic equation are complex conjugates of each other. What can this quartic equation be? What are its roots?
Look up any words you don't understand :)
http://www.analyzemath.com/complex/complex_numbers.html