AAT-A Date: 11/14/13 SWBAT divide polynomials.

Do Now: ACT Prep Problems

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Simple Division -dividing a polynomial by a monomial

2 2 2 26 3 93

r s rs r s1.

rs

6r 2s2

3rs

3rs2

3rs

9r 2s3rs

2rs s 3r

Simplify

3a2b3ab

6a3b2

3ab

18ab3ab

a 2a2b 6

2 3 23 6 183

a b a b ab2.

ab

Simplify

12x2 y

3x

3x3x

4xy 1

212 33

x y x3.

x

Long Division -divide a polynomial by a polynomial

•Think back to long division from 3rd grade.•How many times does the divisor go into the dividend? Put that number on top.•Multiply that number by the divisor and put the result under the dividend.•Subtract and bring down the next number in the dividend. Repeat until you have used all the numbers in the dividend.

-( )x 3 x 2 5x 24

x

x2 + 3x- 8x

- 8

- 24- 8x - 24

0-( )

x2/x = x

2 5 243

x x4.

x

-8x/x = -8

-( )

-( )

-( )h 4 h3 0h2 11h 28

h2

h3 - 4h2

4h2

+ 4h

- 11h4h2- 16h

5h

h3/h = h2

13 11 28 45. h h h

+ 284h2/h = 4h

+ 5

5h - 2048

48

h 4

5h/h = 5

Synthetic Division -

4 26 : 5 4 6 ( 3)Ex x x x x

To use synthetic division: •There must be a coefficient for every possible power of the variable.•The divisor must have a leading coefficient of 1.

divide a polynomial by a polynomial

Step #1: Write the terms of the polynomial so the degrees are in descending order.

5x4 0x3 4x2 x 6Since the numerator does not contain all the powers of x, you must include a 0 for the x3.

4 2

5 4 6 ( 3)x x x x

Step #2: Write the constant r of the divisor x-r to the left and write down the coefficients.

Since the divisor is x-3, r=3

5x4 0x3 4x2 x 6

5 0 -4 1 63

4 25 4 6 ( 3)x x x x

5

Step #3: Bring down the first coefficient, 5.

3 5 0 - 4 1 6

4 25 4 6 ( 3) x x x x

5

3 5 0 - 4 1 6

Step #4: Multiply the first coefficient by r, so 3 5 15

and place under the second coefficient then add.

15

15

4 2

5 4 6 ( 3)x x x x

5

3 5 0 - 4 1 6

15

15

Step #5: Repeat process multiplying the sum, 15, by r; and place this number under the next coefficient, then add.

15 3 45

45

41

4 2

5 4 6 ( 3)x x x x

5

3 5 0 - 4 1 6

15

15 45

41

Step #5 cont.: Repeat the same procedure.

123

124

372

378

Where did 123 and 372 come from?

4 2

5 4 6 ( 3)x x x x

Step #6: Write the 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.

5

3 5 0 - 4 1 6

15

15 45

41

123

124

372

378

4 2

5 4 6 ( 3)x x x x

The quotient is:

5x3 15x2 41x 124 378

x 3

Remember to place the remainder over the divisor.

4 2

5 4 6 ( 3)x x x x

5x 5 21x4 3x3 4x2 2x 2 x 4 Ex 7:

Step#1: Powers are all accounted for and in descending order.

Step#2: Identify r in the divisor.

Since the divisor is x+4, r=-4 .

4 5 21 3 4 2 2

Step#3: Bring down the 1st coefficient. Step#4: Multiply and add.

4 5 21 3 4 2 2

-5

Step#5: Repeat.

20 4 -4 0 8-1 1 0 -2 10

4 3 2 105 24

x x xx

5 4 3 25 21 3 4 2 2 4x x x x x x

6x2 2x 4 2x 3 Ex 8:

6x2

2

2x2

42

2x2

32

Notice the leading coefficient of the divisor is 2 not 1.

We must divide everything by 2 to change the coefficient to a 1.

3x2 x 2 x 32

32

3 1 2

3

92

2

2

72

214

8

4

294

26 2 4 2 3x x x

297 43

322

xx

*Remember we cannot have complex fractions - we must simplify.

7 29 13

32 42

xx

7 293

32 42

xx

7 29

32 4 6

xx

26 2 4 2 3x x x

x3 x 2 2x 7 2x 1 Ex 9:

3 2 2 7 2 12 2 2 2 2 2x x x x

112

12

72

Coefficients

1 1 1 7 1

2 2 2 2

1

2

1

41

4

2

4

1

8

8

8

7

8

7

16

56

16

49

16

3 21 1 7 12 2 2 2

x x x x

3 2 2 7 2 1x x x x

Divide a polynomial by a monomial

Divide a polynomial by a monomial

Slide 2- 26

Steps for Long Division1. Check2. Multiply3. Subtract4. Bring Down

Two Examples

Steps for Long Division1. Check2. Multiply3. Subtract4. Bring Down

Divide a polynomial by a monomial

Rules of Exponents (Keep same base)

1. ax ∙ ay = ax+y Product of powers; add exponents.2. (ax )y = ax y ∙ Power of a power; add exponents.3. (ab)x= ax bx Power of a product ; Distribute exponent to

each term and multiply.4. (a)x= ax – y Quotient of powers, subtract the exponents. (a)y a cannot equal zero

5. Power of a Quotient b cannot equal 0

6. Zero Exponent 7. Negative Exponents (a)0 = 1 a-x = 1 ax

m

mm

ba

ba

x x

x

Scientific Notation: Way to represent VERY LARGE numbers.Standard Notation: Decimal Form

Scientific Notation:

Rules for Multiplication in Scientific Notation: 1) Multiply the coefficients 2) Add the exponents (base 10 remains)

Example 1: (3 x 104)(2x 105) = 6 x 109

Rules for Division in Scientific Notation:

1) Divide the coefficients 2) Subtract the exponents (base 10 remains) Example 1: (6 x 106) / (2 x 103) = 3 x 103

Exit Ticket3rd PeriodPg 428 #4-14 evens

5th/6th pg 428 #8-15

Scientific Notation:

http://ostermiller.org/calc/calculator.html

pg 428 #4-7

Notes: Quotient of Powers:(a)m= ∙ am - n

(a)n To divide powers, keep the same base, subtract the exponents. an cannot equal zero

Zero Exponent(a)0 = 1 a

Negative Exponents Power of a Quotienta-n = 1 a For any integer m and any an real numbers a and b, b

`(

m

mm

ba

ba

0

0

0