WELCOME!!! Warm Up Turn in your polynomial adding and subtracting worksheet Take a piece of paper...
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Transcript of WELCOME!!! Warm Up Turn in your polynomial adding and subtracting worksheet Take a piece of paper...
WELCOME!!! Warm Up
• Turn in your polynomial adding and subtracting worksheet
• Take a piece of paper and fold it into thirds to make a name card. Put your name on the card.
• Divide using long division. No calculator! ______________
32 )435028358602
Video – Not the best, but why not?
Synthetic Division: http://www.youtube.com/watch?v=Ar10yTC9W8M
Objectives
• Multiply polynomials using distributive property • Multiply two binomials using the FOIL method• Multiply polynomials using the box method• Use the Binomial Theorem (Pascals Triangle) to
expand binomials raised to powers• Divide polynomials using long division and
synthetic division• Apply these processes to abstract problems
Today:
• Multiplication Lesson• Multiplication Worksheet• Pascals Triangle Worksheet• Long Division and Synthetic Division Lesson• Division Worksheet• Game• Review• Post Assessment
Do you agree…?3(4+4) = 3(8) = 243(4+4) = 3(4) + 3(4) = 12 + 12 = 24
Why does this happen?Vocab: D_ _ _ _ _ _ _ _ _
Remember this… we will use it very soon!
Multiplying Monomials
Multiply coefficients with coefficients (number next to the variables) and powers of x with powers of x [remember we add exponents when we multiply numbers with same base…. Why? X3X4 = (x x x) (x x x x) = (x x x x x x x) = x7 ]
Try: (4x3)(2x) (remember x is the same as x1)
Try: (15x100)(10x300)
Independent Practice – in your notes
1. (4x)(5x)=
2. (-3x)(8x2)=
3. (12x)(2x3)=
4. (8x)(2x2)(2x4)=
Independent Practice – Answers
1. (4x)(5x)= 20x2
2. (-3x)(8x2)= -24x3
3. (12x)(2x3)= 24x4
4. (8x)(2x2)(2x4)= 32x7
Distributive Property and Polynomials
• Distributive Property: a(b + c) = ab + ac
Independent Practice
5. 3(x2 + 2x + 9) =
6. 4x(x3 + x - 100) =
7. -10x3(2x + 8) =
8. 20x3(4x2 + 8x + 6) =
Independent Practice - Answers
5. 3(x2 + 2x + 9) = 3x2 + 6x + 27
6. 4x(x3 + x - 100) = 4x4 + 4x2 - 400x
7. -10x3(2x + 8) = -20x4 - 80x3
8. 20x3(4x2 + 8x + 6) = 80x5 + 160x4 + 120x3
Multiplying Polynomials
Step 1) Distribute monomials of first polynomial through each term of the second polynomial
Step 2) Multiply the new terms outStep 3) Combine like termsStep 4) Order the terms into Standard Form
FOIL Method
• When multiplying 2 binomials you can FOIL to remember how to distribute
F Firsts / FrontsO OutsideI InsideL Last
(y + 3)(y + 7). F tells you to multiply the FIRST terms
of each binomial.
y2
(y + 3)(y + 7). O tells you to multiply the OUTER
terms of each binomial.
y2 + 7y
(y + 3)(y + 7). I tells you to multiply the INNER
terms of each binomial.
y2 + 7y + 3y
(y + 3)(y + 7). L tells you to multiply the LAST terms
of each binomial.y2 + 7y + 3y + 21
Combine like terms.y2 + 10y + 21
Box Method
Independent Practice
9. (4x+2)(2x+3) =
10. (5x2 – 3)(2x+8)
11. (2x+5) (2x – 5)
12. (3x+5)(3x+5)
Independent Practice - Answers
9. (4x+2)(2x+3) = 8x2 + 12x + 4x + 6 = 8x2 + 16x +6
10. (5x2 – 3)(2x+8) = 10x3 +40x2 - 6x -24
11. (2x+5) (2x – 5) = 4x2 - 25
12.(3x+5)(3x+5) = 9x2 + 15x + 15x + 25 = 9x2 +30x +25
Multiplying Worksheet
Pascals Triangle
? ? ? ?
Binomial Theorem
• You can expand any (x+y)n using the binomial theorem
(x+y)2 = x2 + 2xy + y2 (how many terms do we have?)
(x+y)3 = x3 + 3x2y + 3xy2 + y3 (how many terms do we have?)
Do you see any patterns? How does this relate to pascal’s triangle? Begin working on Exploratory Worksheet
Special Patterns?
(x + y)2 = x2 + 2xy + y2
(x - y)2 = x2 - 2xy + y2
(x + y)(x - y) = x2 - y2
(x + a)(x + b) = x2 + (a + b)x + ab
(x + y)3 = x3 + 3 x2y + 3xy2 + y3 (x - y)3 = x3 - 3x2y + 3xy2 – y3
Long Division
• Do we remember the steps from the warm up earlier?
• Can we apply them using polynomial terms?
Long Division Polynomial Problem
Set it up:
Check!
How did we do?
Independent Practice
13. (m3 – 20) / (m – 3)
14. (5x3 + 11x2 + 26x + 26) / (5x + 6)
Independent Practice - Solutions
13. (m3 – 20) / (m – 3)
m2 +3m + 9 +
14.(5x3 + 11x2 + 26x + 26) / (5x + 6)
x2 + x + 4 +
7 M-3
25x+6
Synthetic Division
• Synthetic Division is an algorithm for when we already know a factor
• We can use this method to divide by binomials of the form (x-a)
• Its important to note that the factor must be
(x-a)
• Examples: (x-2), (x+5), (x - 100) NOT (2x-4) , (4x-5), or (20x + 3)
Long Division ProblemSet it up:
The number that would solve the divisor to zero
All the coefficients of the polynomial dividend terms from highest to lowest (standard form)
NOTE: make sure to include any terms that may have been omitted: 0x2 or 0x wouldn’t show up in a polynomial of standard form, but we need it as a placeholder
Long Division Problem Work
How did we do?
Division Worksheet
Game Preparation
• Take 1 blank piece of paper: Fold your paper once hot-dog and in thirds hamburger creating 6 boxes on your page.
• Number the boxes 1-6
1 2
3 4
5 6
Game Rules/ Procedures
• We will do 6 polynomial review questions• This is an independent game, everyone participates.
You will turn final page in for grade.• Once a questions is flashed, we will begin the timers. • Points will be awarded for the right answer in the
quickest time.• A wrong answer earns you no points, but you are still
required to complete the problem for the final submission after the game!
• (Do your best at your own pace for best results)
Sample Question Format
1. What class is this?2. What units have we covered?3. Y/N Ms. Justin’s Videos were awesome!?4. Am I ready to play?
:30 1:00 1:30 2:00
5pts 4pts 3pts1pt
Question 1:Give an example of….
1. cubic, trinomial2. quadratic binomial3. constant, monomial4. polynomial
:30 1:00 1:30 2:00
5pts 4pts 3pts1pt
Question 2:
:30 1:00 1:30 2:00 5pts 4pts3pts 1pt
1. In words, explain how to subtract polynomial2. (3x3 + 4x2 – 5x + 8) – (2x3 – 8x2 + 4x + 5)
Question 3:
:30 1:00 1:30 2:00 5pts 4pts3pts 1pt
1. What does distribute mean?2. Write as a polynomial in standard form:
5x (4x2 – 5x + 3x3+ 8)
Question 4:
:30 1:00 1:30 2:00 5pts 4pts3pts 1pt
1. What does F O I L stand for?2. What kind of polynomials can we FOIL?3. Find the area of the rectangle:
(3x
+ 8)
(4x2- 5)
Question 5:
• What is this pattern?• What comes next?• How do we use this
when multiplying polynomials?
11 1
1 2 11 3 3 1
:30 1:00 1:30 2:00 5pts 4pts3pts 1pt
Question 6
Divide
1:00 2:00 4:00 6:00
10pts 8pts 5pt2pt
GAME OVER!WHO WON?
HAVE WE MET THE OBJECTIVES?
ARE WE READY FOR THAT ASSESSMENT?
LAST QUESTIONS?
5 MINUTES TO STUDY:
• Polynomial vocabulary• Adding and Subtracting• Multiplying– Distributing, FOIL, special patterns, Pascal
• Dividing– Long, synthetic
POST ASSESSMENT
GOODLUCK