drill #17 simplify each expression.. drill #18 simplify each expression

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Drill #17 Simplify each expression. ) ( ) ( . 3 ) 3 5 ( ) ( . 2 ) 3 3 ( ) 2 3 ( . 1 2 2 2 2 2 z xyz x z yz xyz y x xy xy xy y x x x x x

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Page 1: Drill #17 Simplify each expression.. Drill #18 Simplify each expression

Drill #17Simplify each expression.

)()(.3

)35()(.2

)33()23(.1

222

22

zxyzxzyzxyz

yxxyxyxyyx

xxxx

Page 2: Drill #17 Simplify each expression.. Drill #18 Simplify each expression

Drill #18Simplify each expression.

2

32

322

24

410

224232

3

2.4

)4()3(.3

2

6.2

)3()(.1

ba

ab

xyxy

yx

yx

baba

Page 3: Drill #17 Simplify each expression.. Drill #18 Simplify each expression

Drill #19Simplify each expression.

2

4

3

2322

225142

24

43

2

5.4

)6()3(.3

)4()2(.2

4

10.1

ab

ba

zxyxyz

baba

yx

yx

Page 4: Drill #17 Simplify each expression.. Drill #18 Simplify each expression

Drill #20Simplify each expression.

2

42

5

2312

3

4.2

)4()2

1(.1

ba

ab

xyzzxy

)235(2.4

)2

13()4

2

13(.3

2222

2222

xyxyyxxyxyyx

yxyxyxyx

Page 5: Drill #17 Simplify each expression.. Drill #18 Simplify each expression

Drill #21Simplify each expression.

)2)(2(.3

)2

13

3

1(2

5

1

6

5.2

)3

2(

4

3.1

2222

2321

32

yxyx

yxyxyxyx

zyxz

xy

Page 6: Drill #17 Simplify each expression.. Drill #18 Simplify each expression

Drill #22Simplify each expression.

)32)(7(.3

)2

1

3

1(2

3

2.2

)3

2(

4

3.1

2222

2321

22

xx

yxyxyxyx

zyxz

xy

Page 7: Drill #17 Simplify each expression.. Drill #18 Simplify each expression

Drill #23Simplify each expression.

)1)(3)(7(.3

)4

1

2

1

6

1(2

3

2.2

)4(8.1

322232

232122

xxx

yyzyyz

zyxxy

Page 8: Drill #17 Simplify each expression.. Drill #18 Simplify each expression

Drill #24Simplify each expression.

22221

2

2

2212

)(:

)2)(1(.3

)52(.2

)6

52

2

1(3.1

xyyxyxB

xx

x

yxxyyxxy

Page 9: Drill #17 Simplify each expression.. Drill #18 Simplify each expression

Drill #18

Simplify each expression. State the degree and coefficient of each simplified expression:

)3)(4

1(.3

)4)(3)(2(.2

))(3(.1

2332

3

cbacab

rstrsr

xx

Page 10: Drill #17 Simplify each expression.. Drill #18 Simplify each expression

6-1 Operations With Polynomials

Objective: To multiply and divide monomials, to multiply polynomials, and to add and subtract polynomial expressions.

Page 11: Drill #17 Simplify each expression.. Drill #18 Simplify each expression

Negative Exponents *

For any real number a and integer n,

Examples:

nn

aa

aa

1

11

25

15 2

Page 12: Drill #17 Simplify each expression.. Drill #18 Simplify each expression

Example: Negative Exponent *

2

3

1

1

3

2.

10.

3

2.

)4

1.(

D

C

B

A

Page 13: Drill #17 Simplify each expression.. Drill #18 Simplify each expression

Product of Powers *

For any real number a and integers m and n,

Examples:

nmnm aaa

23535

83535 10101010

aaaa

Page 14: Drill #17 Simplify each expression.. Drill #18 Simplify each expression

Example: Product of Powers*

332

32

34

.

.

.

yxxyC

xxB

ssA

Page 15: Drill #17 Simplify each expression.. Drill #18 Simplify each expression

Quotient of Powers *

For any real number a and integers m and n,

Examples:

nmn

m

aa

a

8)3(53

5

2353

5

101010

10

aaa

a

Page 16: Drill #17 Simplify each expression.. Drill #18 Simplify each expression

Example: Power of a Power*

2

2

53

34

4

3.

3

2.

)(.

)(.

xyD

y

xC

xyB

sA

Page 17: Drill #17 Simplify each expression.. Drill #18 Simplify each expression

Power of a Power*

If m and n are integers and a and b are real numbers:

Example:

mnnm aa )(

6)3(232 )( xxx

Page 18: Drill #17 Simplify each expression.. Drill #18 Simplify each expression

Example: Power of a Power*

2

2

53

34

4

3.

3

2.

)(.

)(.

xyD

y

xC

xyB

sA

Page 19: Drill #17 Simplify each expression.. Drill #18 Simplify each expression

Power of a Product*

If m and n are integers and a and b are real numbers:

Example:

mmm baab )(

333)( yxxy

Page 20: Drill #17 Simplify each expression.. Drill #18 Simplify each expression

Example: Power of a Product*

2321

232

53

)4)(3(.

2

1.

)(.

yxxyC

yxB

xyA

Page 21: Drill #17 Simplify each expression.. Drill #18 Simplify each expression

Power Examples*

Ex1:

Ex2:

Ex3:

223 )2

1()2( aba

34322 )9()3

1( zyxxyz

22 )2

1()

3

2( xyxy

Page 22: Drill #17 Simplify each expression.. Drill #18 Simplify each expression

Find the value of r

Find the value of r that makes each statement true:

1222

42

24

)(

)(

)(

aa

a

xx

x

yy

r

r

r

Page 23: Drill #17 Simplify each expression.. Drill #18 Simplify each expression

Find the value of r *

Find the value of r that makes each statement true:

162.

)(.

)(.

)(.

3

1232

93

2

243

r

r

r

r

D

aa

aC

xx

xB

yyA

Page 24: Drill #17 Simplify each expression.. Drill #18 Simplify each expression

Monomials*

Definition: An expression that is 1) a number, 2) a variable, or 3) the product of one or more numbers or variables.

Constant: Monomial that contains no variables.

Coefficients: The numerical factor of a monomial

Degree: The degree of a monomial is the sum of the exponents of its variables.

Page 25: Drill #17 Simplify each expression.. Drill #18 Simplify each expression

State the degree and coefficient *

Examples:

4

3.

3.

4.

3.

2.

33

5

52

abE

xyzD

zyxC

xyB

yxA

Page 26: Drill #17 Simplify each expression.. Drill #18 Simplify each expression

Polynomial*Definition: A monomial, or a sum (or difference)

of monomials.

Terms: The monomials that make up a polynomial

Binomial: A polynomial with 2 unlike terms.Trinomial: A polynomial with 3 unlike terms

Note: The degree of a polynomial is the degree of the monomial with the greatest degree.

Page 27: Drill #17 Simplify each expression.. Drill #18 Simplify each expression

Polynomials

Determine whether each of the following is a trinomial or binomial…then state the degree:

yxxyyx

xyyx

yzxxyyx

222

3223

222

3

34

3

Page 28: Drill #17 Simplify each expression.. Drill #18 Simplify each expression

Like Terms*

Definition: Monomials that are the same (the same variables to the same power) and differ only in their coefficients.

Examples:

4

3,

3,10

3,23333

5252

abcabc

zyxzyx

yxyx

Page 29: Drill #17 Simplify each expression.. Drill #18 Simplify each expression

Adding Polynomials*

To add like terms add the coefficients of both terms together

Examples

3

2

5 22

abcabc

yxyx

Page 30: Drill #17 Simplify each expression.. Drill #18 Simplify each expression

To combine like terms

To add like terms add the coefficients of both terms together

Example

3

5)3

5()

3

21(

3

2

4)51(5 2222

abcabcabc

abcabc

yxyxyxyx

Page 31: Drill #17 Simplify each expression.. Drill #18 Simplify each expression

Subtracting Polynomials*

To subtract polynomials, first distribute the negative sign to each term in the polynomial you are subtracting. Then follow the rules for adding polynomials. EXAMPLE:

)22()3( yxyx

Page 32: Drill #17 Simplify each expression.. Drill #18 Simplify each expression

Multiplying a Polynomial by a Monomial*

To multiply a polynomial by a monomial:

1. Distribute the monomial to each term in the polynomial.

2. Simplify each term using the rules for monomial multiplication.

)32( 2 xxxy

Page 33: Drill #17 Simplify each expression.. Drill #18 Simplify each expression

FOIL*

Definition: The product of two binomials is the sum of the products of the

F the first terms

O the outside terms

I the inside terms

L the last terms

F O I L

(a + b) (c + d) = ac + ad + bc + bd

Page 34: Drill #17 Simplify each expression.. Drill #18 Simplify each expression

The Distributive Method for Multiplying Polynomials*

Definition: Two multiply two binomials, multiply the first polynomial by each term of the second.

(a + b) (c + d) = c ( a + b ) + d ( a + b )

Page 35: Drill #17 Simplify each expression.. Drill #18 Simplify each expression

Examples: Binomials

)6)(6(:

)5(:

)2)((:

)3)(2(:

2

yyD

xC

xyxB

xxA

Page 36: Drill #17 Simplify each expression.. Drill #18 Simplify each expression

The FOIL Method (for multiplying Polynomials)*

Definition: Two multiply two polynomials, distribute each term in the 1st polynomial to each term in the second.

(a + b) (c + d + e) = (ac + ad + ae) + (bc + bd + be)

Page 37: Drill #17 Simplify each expression.. Drill #18 Simplify each expression

The Distributive Method for Multiplying Polynomials*

Definition: Two multiply two polynomials, multiply the first polynomial by each term of the second.

(a + b) (c + d + e) = c ( a + b ) + d ( a + b ) + e ( a + b )

Page 38: Drill #17 Simplify each expression.. Drill #18 Simplify each expression

Examples: Binomials x Trinomials

3

2

2

)3(:

)1)(5(:

)3)(2)(1(:

)43)(2(:

yD

xxC

xxxB

xxxA

Page 39: Drill #17 Simplify each expression.. Drill #18 Simplify each expression

Classwork: Binomials x Trinomials

3

2

2

)2(:4

)5)(1(:3

)3)(2)(1(:2

)32)(1(:1

y

xx

xxx

xxx

Page 40: Drill #17 Simplify each expression.. Drill #18 Simplify each expression

Pascals Triangle (for expanding polynomials)

1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1