Transcript Document
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