4.1 The Product Rule and Power Rules for Exponents

• Review: PEMDAS (order of operations) – note that exponentiation is number 2.

• Product rule for exponents:

• Example:

mnmn aaa

53232 5555

4.1 The Product Rule and Power Rules for Exponents

• Power Rule (a) for exponents:

• Power Rule (b) for exponents:

• Power Rule (c) for exponents: mmm baab

m

mm

b

a

b

a

nmnm aa

4.1 The Product Rule and Power Rules for Exponents

• A few tricky ones:

16222222

1622222

822222

82222

44

4

33

3

4.1 The Product Rule and Power Rules for Exponents

• Examples (true or false):

222

333

1234

1234

)(

tsts

tsts

tt

ttt

4.2 Integer Exponents and the Quotient Rule

• Definition of a zero exponent:

• Definition of a negative exponent:

is) at matter wha (no 10 a

n

nn

aaa

11

4.2 Integer Exponents and the Quotient Rule

• Changing from negative to positive exponents:

• Quotient rule for exponents:

m

n

n

m

a

b

b

a

nmn

m

aa

a

4.2 Integer Exponents and the Quotient Rule

• Examples:

52

3

2

2

2

2

0

0

22

2

321

110

y

x

y

x

4.3 An Application of Exponents: Scientific Notation

• Writing a number in scientific notation:

1. Move the decimal point to the right of the first non-zero digit.

2. Count the places you moved the decimal point.

3. The number of places that you counted in step 2 is the exponent (without the sign)

4. If your original number (without the sign) was smaller than 1, the exponent is negative. If it was bigger than 1, the exponent is positive

4.3 An Application of Exponents: Scientific Notation

• Converting to scientific notation (examples):

• Converting back – just undo the process:

?

?

102.100012.

102.66200000

000,1861086.1

000,000,000,000,000,000,300,62010203.65

23

4.3 An Application of Exponents: Scientific Notation

• Multiplication with scientific notation:

• Division with scientific notation:

2313

8585

102101021020

101054105104

78412

4

12

4

12

108108.108.

10

10

5

4

105

104

4.4 Adding and Subtracting Polynomials;Graphing Simple Polynomials

• When you read a sentence, it split up into words. There is a space between each word.

• Likewise, a mathematical expression is split up into terms by the +/- sign:

• A term is a number, a variable, or a product or quotient of numbers and variables raised to powers.

35343 22 xyxx

4.4 Adding and Subtracting Polynomials;Graphing Simple Polynomials

• Like terms – terms that have exactly the same variables with exactly the same exponents are like terms:

• To add or subtract polynomials, add or subtract the like terms.

2323 3 and 5 baba

4.4 Adding and Subtracting Polynomials;Graphing Simple Polynomials

• Degree of a term – sum of the exponents on the variables

• Degree of a polynomial – highest degree of any non-zero term

523 degree 5 23 ba

3 degree 100235 23 x xx

4.4 Adding and Subtracting Polynomials;Graphing Simple Polynomials

• Monomial – polynomial with one term

• Binomial - polynomial with two terms

• Trinomial – polynomial with three terms

• Polynomial in x – a term or sum of terms of the form

35x

10035 23 xx

yy 25

xxxaxn 24 3 :examplefor

4.5 Multiplication of Polynomials

• Multiplying a monomial and a polynomial: use the distributive property to find each product.Example:

23

22

2

2012

5434

534

xx

xxx

xx

4.5 Multiplication of Polynomials

• Multiplying two polynomials:

6

2

63x

3

2

2

2

xx

xx

x

x

4.5 Multiplication of Polynomials

• Multiplying binomials using FOIL (First – Inner – Outer - Last):

1. F – multiply the first 2 terms

2. O – multiply the outer 2 terms

3. I – multiply the inner 2 terms

4. L – multiply the last 2 terms

5. Combine like terms

4.6 Special Products

• Squaring binomials:

• Examples:

222

222

2

2

yxyxyx

yxyxyx

11025152515

9633232222

2222

zzzzz

mmmmm

4.6 Special Products

• Product of the sum and difference of 2 terms:

• Example:

22 yxyxyx

222 9333 wwww

4.7 Division of Polynomials

• Dividing a polynomial by a monomial:divide each term by the monomial

555

2

2

2

3

2

23

xx

x

x

x

x

xx

4.7 Division of Polynomials

• Dividing a polynomial by a polynomial:

6

24

84

2

52

24

22854412

2

2

23

2

23

x

x

xx

xx

xx

xxxxxx