40
Drill #29 Simplify each expression. 2 3 4 2 2 4 2 4 4 10 3 2 4 1 2 4 3 2 3 2 . 5 ) 3 ( . 4 2 6 . 3 ) 3 ( . 2 ) 2 )( 3 .( 1 ab b a x y y x y x b a d c ab c b a

Drill #29 Simplify each expression.. Drill #30 Simplify each expression

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Drill #29Simplify each expression.

2

3

4

22

4

24

410

324

12432

3

2.5

)3(.4

2

6.3

)3(.2

)2)(3.(1

ab

ba

x

y

yx

yx

ba

dcabcba

Drill #30Simplify each expression.

2

4

3

2321

22542

24

43

3

2

5.5

)6()3(.4

)3)(2(.3

4

10.2

4

3.1

ab

ba

zxyxyz

baba

yx

yx

Drill #17Simplify each expression.

)()(.3

)35()(.2

)33()23(.1

222

22

zxyzxzyzxyz

yxxyxyxyyx

xxxx

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

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

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

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

Drill #24Simplify each expression.

22221

2

2

2212

)(:

)2)(1(.3

)52(.2

)6

52

2

1(3.1

xyyxyxB

xx

x

yxxyyxxy

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

6-1 Operations With Polynomials

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

Negative Exponents *

For any real number a and integer n,

Examples:

nn

aa

aa

1

11

25

15 2

Example: Negative Exponent *

2

3

1

1

3

2.

10.

3

2.

)4

1.(

D

C

B

A

Product of Powers *

For any real number a and integers m and n,

Examples:

nmnm aaa

23535

83535 10101010

aaaa

Example: Product of Powers*

332

32

34

.

.

.

yxxyC

xxB

ssA

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

Example: Quotient of a Powers*

15

22

5

2

22

2

5

2

10

8.

2

6.

.

.

yx

yxD

x

xC

ba

abB

x

xA

Power of a Power*

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

Example:

mnnm aa )(

6)3(232 )( xxx

Example: Power of a Power*

2

4

62

2

4

5

53

34

4

3.

3

2.

)(.

)(.

x

yxD

y

xC

xyB

sA

Power of a Product*

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

Example:

mmm baab )(

333)( yxxy

Example: Power of a Product*

2321

232

53

)4)(3(.

2

1.

)(.

yxxyC

yxB

xyA

Example: Power of a Product

2

4

62

2

4

5

4

3.

3

2.

x

yxE

y

xD

Power Examples*

Ex1:

Ex2:

Ex3:

223 )2

1()2( aba

34322 )9()3

1( zyxxyz

22 )2

1()

3

2( xyxy

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

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

MonomialsDefinition: An expression that is 1) a number, 2)

a variable, or 3) the product of one or more numbers or variables.

NOTE: variables must have WHOLE number exponents

Constant: Monomial that contains no variables.Coefficients: The numerical factor of a

monomialDegree: The degree of a monomial is the sum of

the exponents of its variables.

State the degree and coefficient *

Examples:

4

3.

3.

4.

3.

2.

33

5

52

abE

xyzD

zyxC

xyB

yxA

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.

Polynomials

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

yxxyyxEx

yxyxyxEx

xxEx

242

22223

25

3:3

34:2

146:1

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

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

Adding Polynomials and Subtracting Polynomials

)53()43(:3

)594()876(:2

)132()654(:1

22

22

22

xxxxex

xxxxex

xxxxex

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.

)1296(3

4:2

)32(:1

22

2

xxxEx

xxxyEx

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

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 )

Examples: Binomials

)6)(6(:

)5(:

)2)((:

)3)(2(:

2

yyD

xC

xyxB

xxA

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)

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 )

Examples: Binomials x Trinomials

3

2

2

)3(:

)1)(5(:

)3)(2)(1(:

)43)(2(:

yD

xxC

xxxB

xxxA

Classwork: Binomials x Trinomials

3

2

2

)2(:4

)5)(1(:3

)3)(2)(1(:2

)32)(1(:1

y

xx

xxx

xxx

Pascals Triangle (for expanding polynomials)

1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1