Monomials An expression that is either a number, a variable, or a product of numerals and variables with whole number exponents.

Monomials 5x 3 y 12 is a monomial 2y

x

2 is not a monomial 5x y z 1 is not a monomial

Vocabulary  Constants monomials that contain no variables Example 3 or -22  Coefficient Numeric factor of the term -32x 3 y 12 z 15 coefficient = -32

Vocabulary (continued)  Degree of a Monomial The sum of the exponents of the variables 3x 4 degree = 4 -32x 3 y 12 z 15 5 degree = 3+12+15 = 30 degree = 0

Vocabulary (continued)  Power An expression in the form of x n Can also refer to the exponent

Product of Powers  For any real number a and integers m and n, a m · a n =a m+n 2 3 · 2 5 =2 · 2 · 2 · 2 · 2 · 2 · 2 · 2= 2 8

Quotient of Powers  For any real number a and integers m and n,

a a m n



a

5 5 5 2      5 5  3

Quotient of Powers  Find the quotient 2 3 2 8  2  5 2 3 2 8  2  5        1 5

NEGATIVE EXPONENTS  For any real number a≠0 and any integer n, a -n =

a

1

n

2  3

b

1  8  1 2 3 

b

8  1 8

Vocabulary (continued) Simplify rewrite expression  No parenthesis  No negative exponents  Multiply variables  Combine like terms

Simplify (-2a 3 b)(-5ab 4 )  Multiply Coefficients (-2)(-5)=10  Multiply Variables (a 3 )(a) = a 4 (b)(b 4 ) = b 5  10a 4 b 5

 Try this one Simplify 14 21

a b c a bc

 3 2 3

b c a

3 9 2

PROPERTIES OF POWERS • Power of a Power: (a m ) n =a mn • Power of a Product: (ab) m =a m b m • Power of a Quotient:

a b n n

 

n



b n a n

Properties of Powers   4 

b

8 

b

 2a

b

2  5    2a   5  5   32a

b

10 5  4 3

x

4  3

x

4 4  81

x

4

Scientific Notation FORM a x 10 n 1 10 n is an integer Write in Scientific Notation 4,560,000 0.000092

Multiply Numbers in Scientific Notation (a x 10 n ) (b x 10 m ) = (ab x 10 n+m ) Check and make sure 1 

ab

 10 (1.8 x 10 4 ) (4 x 10 7 ) (5 x 10 3 ) (7 x 10 8 )

Divide Numbers in Scientific Notation

a b

  10

m

10

n

10 Check and make sure 1 10   10 6

Polynomials A monomial or a sum of monomials.

Monomial – a polynomial with exactly one term Binomial – a polynomial with exactly two terms Trinomial – a polynomial with exactly three terms

Polynomial Vocabulary  Term Each monomial in a polynomial  Like Terms Terms whose variable factors are exactly the same  Degree of the Polynomial The highest degree of its terms

Polynomials • Indicate if the following is a polynomial, • If so classify according to the number of terms • Indicate the degree of the polynomial

c

4  4  16

p

5 

c

3 4  18 Not a polynomial Polynomial- Binomial- 9

Simplify (2a 3 +5a-7) + (a 3 -3a+2)  3a 3 +2a-5 (3b 3 +2b 2 -4b+3) - (b 3 -2b 2 +3b-4)  2b 3 +4b 2 -7b+7 -3y(4y 2 +2y-3)  -12y 3 - 6y 2 + 9y

Polynomial Vocabulary (continued)  Leading Term The term with the highest degree  Leading Coefficient The coefficient of the leading term

Descending Order  A polynomial is written in descending order for the variable x when the term with the greatest exponent for x is first, and each subsequent term has an exponent for x less than the prior term.

 Example: Write the following in descending order for the variable a. 4a 4 + a 2 - 7a 3 +6a 5 + 12a 8 + 4 12a 8 + 6a 5 + 4a 4 - 7a 3 + a 2 + 4

Multiplying Polynomials  2

p



p

 1  8

p

2  14

p

 3

Multiplying Polynomials 

a

2  3

a

  

a

 1  2

a

3  5

a

2  11

a

 4