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