Lesson 4.2 Multiply & Divide Polynomials Objectives: • Use BOX method to Multiply Polynomials • Use Synthetic Division to Divide Polynomials Multiply & Divide Polynomials

2 X 2 for a Binomial X Binomial Concept: Using the Box method Multiply: (x + 2)(x + 4) Step 1: Draw the box. (What are the dimensions?) 2 X 2 for a Binomial X Binomial

Concept: Using the Box method Multiply: (x + 2)(x + 4) Step 2: Set up the box I’m going to need a lot more room than this…

Concept: Using the Box method Multiply: (x + 2)(x + 4) Step 3: Multiply row by column. ………………………………………………. This part is kind of like filling in a multiplication times table.

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Concept: Using the Box method Multiply: (x + 2)(x + 4) Step 4: Combine like terms. ……………………………….. You might notice that the “like” terms kind of line up with each other diagonally.

You Rock

That’s amazing. LET’S DO THAT AGAIN! FASTER!!!

2x2 for a Binomial x Binomial Concept: Do it Again Hold on! Here we go!!! 2x2 for a Binomial x Binomial

YOU TRY What Do I Do First?????

2x2 for a Binomial x Binomial

Concept: Other Sized Polynomials Does the Box Method work with any size? YES! This is a binomial times a trinomial. What will the dimensions of the box be? 2 X 3

Concept: Other Sizes cont… 2x2 +10x –6 10x3 50x2 –30x 30x 6x2 –18 5x +3 10x3 + 50x2 + 30x + 6x2 – 30x – 18 10x3 + 56x2 – 18

Concept: You Try!!! Multiply. 1. (x + 2)(x – 8) x2 – 6x – 16 3. (2x – 5y)(3x + y) x2 – 6x – 16 2x3 – x2 – 29x + 28 6x2 – 13xy – 5y2

Long Division----Yuck!!!

Concept: Division Requirements In order to use Synthetic Division there are two things that must happen: #1 There must be a coefficient for every possible power of the variable. #2 The divisor must have a leading coefficient of 1. Ex: . Given

Concept: Setting It Up Ex: . Given Step #1: Start out EXACTLY like you did when you were going to evaluate.

Concept: Setting It Up cont. . . Write the opposite of the constant (a) of the divisor to the left and write down the coefficients. (NOTE: THIS PART IS NEW. ) Step #2: Ex:

Concept: Dividing cont… Step #3: From this point on it is EXACTLY the …............ the same as the evaluating in 4.1

Concept: Writing the answer • The answer to a division problem is called a quotient. The numbers along the bottom are coefficients of the power of x in descending order, starting with the power that is one less than that of the dividend. If the sum of the numbers in the last column is NOT zero, it is a remainder and will be written as the numerator of a fraction. • •

Remember to place the remainder over the divisor. Concept: Writing the answer cont… Given: The bottom row of numbers in the division problem was: ….... 5 15 41 124 378 The quotient is: Remember to place the remainder over the divisor.

Concept: You Try!!!

Divide: f (x )= (2x 3 + x 2 – 8x + 5) ÷ (x + 3) Concept: You Try!!! Divide: f (x )= (2x 3 + x 2 – 8x + 5) ÷ (x + 3) SOLUTION: –3 2 1 –8 5 –6 15 –21 2 –5 7 –16 = 2x 2 – 5x + 7 – 16 x + 3 ANSWER:

Concept: You Try!!!

Concept: Divisor LC ≠ 1 Ex: When the leading coefficient of the divisor is not 1, you must: • Set the divisor = 0. • Solve for x. (This number will go in the box) Ex: 2x –1 = 0 2x = 1

Concept: Divisor LC ≠ 1 cont… Ex: 8 _____ Now, just do the division just like before!!!

Concept: Divisor LC ≠ 1 cont… Ex: 8 7 1 –5 –8 ______________________________ 14 2 –10 –16

Concept: You Try! Divide: HINT: Keep changing your calculations back into fractions after each step to keep the answer exact. Sometimes this gets a bit messy.

Homework: PM 4.2 PM 4.2