Monomials

An expression that is either a number, a variable, or a product of numerals and

variables with whole number exponents.

Monomials

5x3y12 is a monomial

is not a monomial

is not a monomial

2

2yx

13 12 35x y z

Vocabulary

Constants monomials that contain no variables

Example 3 or -22

Coefficient Numeric factor of the term

-32x3y12z15 coefficient = -32

Vocabulary (continued)

Degree of a MonomialThe sum of the exponents of the variables3x4 degree = 4-32x3y12z15 degree = 3+12+15

= 305 degree = 0

Vocabulary (continued)

PowerAn expression in the form of

xn

Can also refer to the exponent

Product of Powers

For any real number a and integers m and n,

am · an =am+n

23 · 25 =2 · 2 · 2 · 2 · 2 · 2 · 2 · 2= 28

Quotient of Powers

For any real number a and integers m and n,

mm n

n

aa

a

53

2

5 5 5 5 5 55

5 5 5

Quotient of Powers

Find the quotient

5

2 2 2 12 2 2 2 2 2 2 2 2

3

5

8

22

2

35

8

22

2

NEGATIVE EXPONENTS

For any real number a≠0 and any integer n, a-n=1

na

33

88

1 12

2 81

bb

Vocabulary (continued)

Simplifyrewrite expression

No parenthesisNo negative exponentsMultiply variablesCombine like terms

Simplify

(-2a3b)(-5ab4)Multiply Coefficients

(-2)(-5)=10Multiply Variables

(a3)(a) = a4

(b)(b4) = b5

10a4b5

Simplify

Try this one

3 4 6

5 3

1421a b ca bc

3 9

2

23b ca

PROPERTIES OF POWERS

• Power of a Power: (am)n=amn

• Power of a Product: (ab)m=ambm

• Power of a Quotient: n n

n

a ab b

nn n

n

a b bb a a

Properties of Powers

4

3x

2 4b

5

52

2a

b

42b

5

2

2ab

8b

5

10

32ab

4

4

3x

43

x 4

81x

Scientific Notation

FORM a x 10n

n is an integer

Write in Scientific Notation4,560,0000.000092

1 10a

Multiply Numbers inScientific Notation

(a x 10n) (b x 10m) = (ab x 10n+m)Check and make sure

(1.8 x 104) (4 x 107)(5 x 103) (7 x 108)

1 10ab

Divide Numbers inScientific Notation

Check and make sure

1010

10

mm n

n

aa b

b

1 10a b

6

10

2.7 109 10

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 TermsTerms 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

4

5 2 7

4 18

316

4

c c

p p q

Not a polynomial

Polynomial- Binomial- 9

Simplify

(2a3+5a-7) + (a3-3a+2)3a3+2a-5(3b3+2b2-4b+3) - (b3-2b2+3b-4)2b3+4b2-7b+7-3y(4y2+2y-3)-12y3 - 6y2 + 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.

4a4 + a2 - 7a3 +6a5 + 12a8 + 412a8 + 6a5 + 4a4 - 7a3 + a2 + 4

Multiplying Polynomials

2 3 4 1p p

28 14 3p p

Multiplying Polynomials

2 3 4 2 1a a a

3 22 5 11 4a a a