Download Document

Algebra Notes 8-1 Multiplying Monomials
A. Multiplying Monomials
1. Definitions
a. A monomial is a number, a variable, or a
product of a number or variable.
Ex 1: Determine if each expression is a monomial.
a.) 17 − s
b.) 8 f 2 g
c.)
3
4
d.) xy
2. To multiply monomials, write everything out.
Ex 2: Simplify each expression
a.) (2r 4 )( −12r 5 )
b.) (6cd 5 )(5c5 d 2 )
Ex 3: Simplify each expression
a.) ( x 2 y )3
b.) [(a 3b 4 ) 2 ]3
Ex 4: Find the area of a square if a side is 5xy.
Ex 5: Simplify [(8 g 3h 4 ) 2 ]2 (2 gh5 ) 4
HW: Algebra 8-1 p. 413-415
15-39 odd, 43-46, 59-60, 63-65, 67-68, 75-82
8-2 Dividing by Monomials
Ex 1: Simplify the following.
y4 z2
a.) 2 =
y z
2
 4c d 
b.)  2 4  =
 5e f 
3
2
8 x 4 y10 a 3b
c.)
=
12 2
2 xy b
1. If you have a −2 , then you must switch the
1
numerator and the denominator, so a −2 = 2
a
2. Any number raised to the zero power is 1.
Ex 2: Simplify the following.
x −6
a.) −4 −5 =
y z
75 p 3 q −5
b.)
=
5 0 −8
15 p q r
y −2 x 4 z −5
c.) −1 0 −6 =
x y z
HW: Algebra 8-2 p. 421-423
15-37 odd, 38, 43, 47-48, 55-59 odd, 63-77 odd,
8-3 Scientific Notation
A. Scientific Notation
1. Very large or very small numbers can be written
more easily in scientific notation.
2. The number is written a × 10n where a is at least
one but less than ten, and n is an integer.
(1 ≤ a < 10 )
Ex 1: Write the numbers in standard notation.
a.) 7.23 × 105
b.) 5.17 × 10−4
Ex 2: Write the number in scientific notation.
a.) 507,000,000
b.) 0.0000123
3. A lot of times you can multiply or divide numbers
in scientific notation easily.
(2.2 ×103 )(3 ×105 ) = 2.2(3) × 103 (105 )
Ex 3: Evaluate - answer in both scientific and
standard notation.
a.) (3 × 1012 )(3 × 10−6 )
6.4 × 103
b.)
1.6 × 106
HW: Algebra 8-3 p. 428-430
19-41 odd, 45-55 odd, 62-63, 64-73, 77-82
8-4 Polynomials
A. Definitions
1. A polynomial is monomial or sum of
monomials.
2. A binomial is two monomials added together.
3. A trinomial is three monomials added
together.
Ex 1: State whether the expression is a
polynomial, if it is, state whether it is a monomial,
binomial, or trinomial.
2z
a.) 7 x 2 yz 3 +
+ 14
xy
b.) 6 − 4
c.) x 2 + 2 xy − 12
d.) 256a 5b5c 6
4. The degree of a monomial is the sum of the
exponents of the variables.
Ex 2: Find the degree of each monomial.
a. 4x 2
b. 3abc
c. −20x 2 y 3 z
5. To find the degree of a polynomial, it is the
degree of the greatest term.
Ex 3: Find the degree of each polynomial.
a. 3 x 2 + 5
b. y 7 + y 6 + 3 x 4 m 4
6. To arrange a polynomial in descending order,
put the powers in order from highest to lowest.
Ex 4: Arrange the x-terms in descending order.
a. y 4 x + y 5 x3 − x 2 + yx5
b. 7 x 2 − x5 − 12 x 4 + 3 + x + 7 x 7
HW: Algebra 8-4 p. 434-436
15-20, 22, 25-36, 45-51 odd, 59-60, 61-71 odd, 72-76
8-5 Adding and Subtracting Polynomials
A. Adding
1. In order to add polynomials, you may only add
like terms.
Ex 1: Simplify
a.) (7 y 2 + 3 y + 5) + (2 y 2 + 4 y + 1)
b.) ( x 3 + 2 x 2 − 4 x + 5) + (2 x 3 + 2 x − 1)
B. Subtracting
1. When you subtract polynomials, you MUST
distribute the minus sign to all the monomials
that follow the subtraction sign.
Ex 2: Simplify
a.) (2a 2 + 5a − 4) − (5a 2 − a + 3)
b.) (8 x 4 − 2 x 2 + 3) − (6 x 2 + 7 x 4 − 3)
Ex 3: The measure of the perimeter of the triangle
shown is 37s + 42.
14 s + 16
a.) Find the third side.
b.) Find the length of the
third side if s = 3 meters.
10 s + 20
HW: Algebra 8-5 p. 441-443
13-29 odd, 30-31, 45-46, 47-53 odd, 60-61, 63-68
8-6 Multiplying a Polynomial by a Monomial
A. When you multiply a polynomial by a
monomial, you must be sure to distribute both the
variable and the coefficient to the terms in the
polynomial.
Ex 1: Simplify
a) 7b(4b 2 − 9) =
b) 2 x(4a 2 + 3 x3 − 5ax) =
c) −3 pq ( p 2 q + 2 p − 3 p 2 q) =
d)
4 2
5
x (9 xy + x − 25 y ) =
5
4
e) 4 y (2 y 3 − 8 y 2 + 2 y + 9) − 3( y 2 + 8 y ) =
f) Solve for x:
1.) −2( w + 1) + w = 7 − 4 w
2.) x( x − 4) + 2 x = x( x + 12) − 7
HW: Algebra 8-6 p. 446-448
15-35 odd, 38, 39-47 odd, 53-54, 64-65, 67-79 odd, 82-87
8-7 Multiplying Polynomials
A. FOIL – to multiply any polynomial, you must
FOIL, multiply:
F – First terms
O – Outer terms
I – Inner terms
L – Last terms
Ex 1: Find each product
a. ( x + 4)( x − 9) =
b. ( a + 5)(2a − 4)
c. ( y − 3) 2 =
d. (2 y + 5)(3 y 2 − 8 y + 7) =
HW: Algebra 8-7 p. 455-457
13-41odd, 45-46, 56-57, 59-71 odd, 72-77
8-8 Special Products
A. Special products
Ex 1: Find each product
a.) ( x + 5) 2
b.) (7 a − 4) 2
c.) (5 y 4 − 2 y ) 2
d.) ( x + 4)( x − 4)
-this result is called a difference of squares.
e.) (11r + 7 s )(11r − 7 s )
HW: Algebra 8-8 p. 462-463
13-37 odd, 41-44, 49-50, 53-67 odd, 70