6.1 Review of the Rules for Exponents • Product rule for exponents:

a n



a m



a n



m

• Power Rule (a) for exponents: • Power Rule (b) for exponents:  

n

 

m



a nm



a m b m

• Power Rule (c) for exponents:

a b m



a m b m

6.1 Review of the Rules for Exponents • Changing from negative to positive exponents:

a



m b



n



b n a m

• Quotient rule for exponents:

a m a n



a m



n

6.2 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: 3

x

 4

x

2  3

xy

2  • A term is a number, a variable, 35 or a product or quotient of numbers and variables raised to powers.

6.2 Adding and Subtracting Polynomials; Graphing Simple Polynomials • Like terms – terms that have exactly the same variables with exactly the same exponents are like terms: 5

a

3

b

2 and  3

a

3

b

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

6.2 Adding and Subtracting Polynomials; Graphing Simple Polynomials • Degree of a term – sum of the exponents on the variables 5

a

3

b

2 degree  3  2  5 • Degree of a polynomial – highest degree of any non-zero term 5

x

3  3

x

2  2

x

 100 degree  3

6.2 Adding and Subtracting Polynomials; Graphing Simple Polynomials • Monomial – polynomial with one term 3 5

x

• Binomial - polynomial with two terms 5

y

2 

y

• Trinomial – polynomial with three terms 5

x

3  3

x

2  100 • Polynomial in x – a term or sum of terms of the form

ax n

for example :

x

4  3

x

2 

x

6.3 Multiplication of Polynomials • Multiplying a monomial and a polynomial: use the distributive property to find each product.

Example: 4 

x

2



3

x

 5



 

 

  

 

 12

x

3  20

x

2

6.3 Multiplication of Polynomials • Multiplying two polynomials:

x

 2

x

 3  3x  6

x

2  2

x x

2 

x

 6

6.3 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

6.3 Multiplication of Polynomials • Squaring binomials:



x



y



2 

x

2 



x



y



2 

x

2  2

xy

 2

xy



y

2

y

2 • Examples:



m

 3



2 

m

2



5

z

  1

  

2 2

 

  3 2 2

 

 1 2  

m

2  25

z

2 6

m

 9  10

z

 1

6.3 Multiplication of Polynomials • Product of the sum and difference of 2 terms: 

x



y x



y

 

x

2 

y

2 • Example:  3 

w

  3 

w

  3 2 

w

2  9 

w

2

6.4 Division of Polynomials • Dividing a polynomial by a monomial: divide each term by the monomial

x

3  5

x

2

x

2 

x

3

x

2  5

x

2

x

2 

x

 5

6.4 Division of Polynomials • Dividing a polynomial by a polynomial: 2

x

2 

x

 2 2

x

 1 4

x

3  4

x

2  5

x

 8 4

x

3  2

x

2  2

x

2  5

x

 2

x

2 

x

4

x

 8 4

x

 2  6

6.4 Division of Polynomials • Synthetic division: 2 1 1  5 2  3 7  6 1

x

3  5

x

2

x

  2 7

x

 3  3 2  1 answer is:

x

2  3

x

 1 remainder is: -1 

x

2  3

x

 1 

x

1  2