Divide using long division. No calculator! 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) = 24 3(4+4) = 3(4) + 3(4) = 12 + 12 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) =

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 out Step 3) Combine like terms Step 4) Order the terms into Standard Form

FOIL Method F Firsts / Fronts O Outside I Inside L Last When multiplying 2 binomials you can FOIL to remember how to distribute F Firsts / Fronts O Outside I Inside L Last

(y + 3)(y + 7). F tells you to multiply the FIRST terms of each binomial.

(y + 3)(y + 7). O tells you to multiply the OUTER terms of each binomial.

(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 (3x+5)(3x+5) = 9x2 + 15x + 15x + 25 = 9x2 +30x +25

Multiplying Worksheet

Pascals Triangle ? ? ? ?

Do you see any patterns? 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 + (5x3 + 11x2 + 26x + 26) / (5x + 6) x2 + x + 4 + 7 M-3 2 5x+6

Synthetic Division (x-a) 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 Problem Set it up: 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 The number that would solve the divisor to zero All the coefficients of the polynomial dividend terms from highest to lowest (standard form)

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 What class is this? What units have we covered? Y/N Ms. Justin’s Videos were awesome!? Am I ready to play? :30 1:00 1:30 2:00 5pts 4pts 3pts 1pt

Question 1: Give an example of…. cubic, trinomial quadratic binomial constant, monomial polynomial :30 1:00 1:30 2:00 5pts 4pts 3pts 1pt

Question 2: In words, explain how to subtract polynomial (3x3 + 4x2 – 5x + 8) – (2x3 – 8x2 + 4x + 5) :30 1:00 1:30 2:00 5pts 4pts 3pts 1pt

Question 3: What does distribute mean? Write as a polynomial in standard form: 5x (4x2 – 5x + 3x3+ 8) :30 1:00 1:30 2:00 5pts 4pts 3pts 1pt

Question 4: What does F O I L stand for? What kind of polynomials can we FOIL? Find the area of the rectangle: (4x2- 5) (3x + 8) :30 1:00 1:30 2:00 5pts 4pts 3pts 1pt

Question 5: 1 1 2 1 1 3 3 1 What is this pattern? What comes next? How do we use this when multiplying polynomials? 1 1 2 1 1 3 3 1 :30 1:00 1:30 2:00 5pts 4pts 3pts 1pt

Question 6 Divide 1:00 2:00 4:00 6:00 10pts 8pts 5pt 2pt

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

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