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Transcript
Polynomials :
Def. A monomial is a number, a variable, or a product of numbers and variables. Exponents on variables
must be nonnegative integers.
ex. 3x, y, - 2, 1/3, 3x2y , ….
ex. ( not a monomial )
x – 2, 3x/y, \/ x3 ,
Def. A polynomial is a variable expression in which the terms are monomials ( a sum of monomials)
A polynomial with one term is called a _______________
with two terms Î _____________
with three terms Î ______________
Addition of polynomials. Combine similar terms ( like terms)
ex. ( 3x2 + 2x – 3 ) + ( 5x2 - 4x - 12 ) = ____________________
ex. ( 2xy + x ) + ( 5xy + y ) = _____________________________
Subtraction:
We perform subtraction operations as before - write as an addition problem and use addition
rules.
ex. ( 3x – 2y – 1 ) - ( 2x – 5y + 2 ) = ___________________________
ex. ( 3x2 - 3 ) - ( 4x - 2 ) = ______________________________
Degree
of a monomial:
x4 Î _________
x3y5 Î ________
4x7 Î ___________
- 3x Î ___________
23 Î __________ 8x6y3 Î _______
Degree of a polynomial:
Find the degree of each of the terms of the polynomial and then select the largest of the degrees.
ex. 3 + 2x + x2 + x4 Î _______________
ex. 2x4y5 + 1 Î _________
8x8y9 + x15 Î ____________
4x8 - 2x + 3 Î ______________
310 + 2x Î ____
101
Multiplication of Polynomials:
Rules of exponents:
1. If m and n are positive integers ( natural numbers ) , then
xm • x n = xm + n
ex. Find a4 • a8 = ___________
43 • 410 = _________
What about x3 • y5 = ___________ or
(-2 )4 (-2)3 = __________
( 4x3 ) ( 3x2 ) = _________________
( 2a3 b ) ( a2b5 ) = ___________
2. If m and n are positive integers, then ( x m ) n = _____________
ex. Find ( 23 )4 = _________
( x3 ) 2 = ______________
3. If m, n, and p are positive integers, then
( xm yn )
p
= _______________
ex ( 2x3 ) 4 = ____________
( x4 y2 ) 5 = ________________
These types of rules help us multiply monomials together but what about the product of polynomials ?
More examples:
2x(3x2 ) = _______
(2xy)(3x)(5y2 ) = ________
-4x5y( 3xy2 ) = __________
- ( y2 ) 2 = _______
2x(x2 – 3y ) = ________
( - y4 )3 = _______ ( 4x2y3 )3 = _____
102
Product of Polynomials.
1. Use the distributive law to multiply a polynomial by a monomial
- 4 ( x2 + 3xy ) = _____________
ex. 3 ( 2x – y ) = ___________
2x2 ( 3x3y – 4x ) = __________
a2b ( 2ab4 + b + 1 ) = _________________
2. A special case of a product of two polynomials: the product of two binomials
We can use the distributive property and come up with what we call the FOIL method
examples.
( x + 2 ) ( x + 5) = ________________
( 2x – 1 ) ( x + 3 ) = ______________
( 3a – 4 )( 2a + 3 ) = ______________
( 2 – 3y ) ( 4 + 2y ) = _____________
3. What would we do to find these products
a) ( x + 3 ) ( 2x – y + 3 ) = __________________
b) ( 3x + 2y) ( x – 2y + 5 ) = __________________
Def. If x ≠ 0 , then x0 = 1. Notice that 00 has no defined value.
Ex. (-3 )0 = ________
but - 30 = ______
- 4r0 = _______
ex. What do you think x0 equals if x = - 2 ? x0 = _________
Def. If n is a positive integer and x ≠ 0, then x
ex.
a) 2-3 = _______
-n
= 1/ xn and 1/ x-n = xn
4 – 1 = _______
3-3 = __________
103
More Rules of Exponents.
4. If m and n are positive integers and x ≠ 0 , then xm / x n = _________
In the event that the result is a negative integer we can use the fact that x-n = 1/ xn.
examples:
b4 / b3 = ________
b5 / b8 = ________
x2y
4x5
-------- = __________
2x8
------------ = __________
x2y3
Scientific Notation:
A number written in the form y.xxxx • 10n is said to be written in Scientific notation.
y must be between 1 and 10 and n is an integer.
Convert to scientific notation
345600 = ______________
300100 = _________________
0.023 = _______________
0.000501 = ________________
Convert to standard form
3.01 x 105 = ____________
4.2 x 10 – 2 = _______________
104
Verbal Expressions – to - Variable Expressions page 347
word phrases to algebraic phrases
Other examples on pag 351: together in class
2) ______________________
6) _____________________
10) _____________________
14) _____________________
18) _____________________
22) _____________________
26) _____________________
30) ______________________
34) _____________________
HW: page 352: even problems: 2 – 66 ( skip the ones we did above)
105
Chapter Six. First Degree Equations
First Degree Equations: one variable , with a power of 1
( We will also look at those with two variables with power of 1- but that will be later)
ex. 3x – 2 = x + 1
2x – 3 ( 1 – x ) = x
2x + 3 = 5
3x = 1/3
Recall the operations that you are allowed to use to solve these equations:
1) add equal quantities to both sides to create an equivalent equation ( same solution)
-- this includes subtraction
2) you can multiply both sides (or divide) by any nonzero quantity and still have an equivalent
equation ( same solution )
ex.
c + 2/3 = ¾
- 5z + 5 + 6z = 12
-72 = 18v
- z/4 = 3
7y - 2/5 = 12/5
3(2z – 5) = 4z + 1
page 379: 72
page 387: 59
We have now reviewed solving equations of different forms.
x+a=b
ax + b = c
ax + bx = c
ax + c = bx + d
x+3=-2
3x – 2 = - 4
4x – 7x = 2
2x – 3 = -3x – 2
Use these ideas to solve word problems.
Word Problems: Translating sentences into equations
106
2/393 The difference between 9 and the number is seven. Find the number
8/393 Six less than four times a number is twenty-two. Find the number.
18/394
The sum of two numbers is twenty-five. The larger number is five less than four times the smaller
number. Find the two numbers.
22/394 The height of a computer monitor screen is 15 in. This is three-fourths the length of the screen.
What is the area of the monitor screen ? ( slight change from text)
30/394 A 14-yard fishing line is cut into two pieces. Three times the length of the longer piece is four
times the length of the shorter piece. Find the length of the each piece.
107
Rectangular Coordinate System:
origin, axes, plane, abscissa(x-coordinate), ordinate(y-coordinate)
Plane: we can identify points on the plane, labeled (x,y)
Plot the points A( 3, 4 ) ________
D( 4, 0 ) _________
B( -2, 3 ) ________
C ( - 4, 1 ) = ________
E( 0, - 2 ) ___________
Scatter Diagrams: graph of ordered pairs of the form (x, y) – relationship between two variables.
See page 399.
Also,
Amount of time ( Study)/ week:
Grade on Exam:
Graph:
108
We have seen linear equations in one variable:
look at linear equations in two variables: 2x – y = 3, y = -4x + 3 ,...
an equation of the form y = x2 + 2x + 1 is not a linear equation, neither is xy + 2x = 3
ex. We say (2, 3 ) is a solution of 2x – y = -1 if x =2, y = 3 make the equation a true statement.
ex. Is ( 3, - 1 ) a solution of x – y = 2 ?
ex. If x = 1 is part of a solution of x – 2y = 2, then what is y ?
An equation of the form y = mx + b is of special interest to us.
If we graph enough equations of this form, we start seeing that these equations represent __________
ex. Sketch the graph of y = 2x + 2
x
y
==========
2
0
-1
3
Graph
y = 2/3 x + 3
109
Name _____________________________ Math 130A – Long Quiz – October 28, 2002
1. If you walk 2 kilometers, then how many meters have you moved ? _________________
2. A slow moving bug travels at 6 cm per second. How many cm will it travel in 1hour ?
___________
How many meters is that ? _________
3. Which unit would be the best to measure the distance between here and Austin
millimeters, meters,
kilometers, decameters
4. Which of the basic units would best be used to measure the amount of coffee in your cup
gram , meter,
liter
5. Graph y = 2x - 4
6. Plot the points A ( -3, 0 ) and B ( 2, - 4 ) . Label the quadrants, label the axes.
110
Review over Coordinate system
1. Construct the rectangular coordinate system
include, axes and quadrants.
2. Use the graph above to plot the following points; A(2, -4 ), B( -2, 0 ), and C( -2, -5 )
3. What quadrants should the point P(x,y) be if
a) x and y are both positive ? __________
b) x is positive and y is negative ? ______
4. Is ( -2, -3 ) a solution of the equation 2x – y = -1 ? SHOW!!
5. What should x be if (x, -2) is a solution of 3x – y = 1 ? x = ______
6. Use another coordinate system to graph each of the following equations
a) 2x – y = 3
b) y = -
−2
x+2
3
111
Chapter 7 The Metric System of Measurement
( How many meters in a )
kilo _________
hecto _______
deca __________
==> ___________
( A meter has )
deci __________
centi _________
kilo
hecto
103
1000
102
100
deca
101
10
milli _________
base-unit
deci
1
1
centi
10-1
1/10
milli
10-2
1/100
10-3
1/1000
distance: meter
examples:
A man walks 300 steps if each step is 1 meter long , then how many kilometers has he walked ?
A piece of paper is 0.32 meters long. How many centimeters is this ?
mass(weight) : gram : 1 x 1 x 1cm3 (mass , weight of water)
examples:
an object weighs 200 g. How many decigrams is this ? _________
A person weighs in at 100 kg. How many grams is this ? ________
capacity(volume): liter ( 10x10x10cm3 )
examples:
A mad scientists requires 32 ml of brain fluid. How many liters is this ? ________
23 liters of gas is equal to how many milliliters of gas ? ________________
Convert from one unit to the other:
See notes from class
112
ratio: comparison of two quantities with the same units – can be written as a fraction, with a colon, in
words
ex. If Joe runs three miles and Ralph runs two miles then write the ratio the distance Joe ran to the
distance Ralph ran.
ex. Kim recipe calls for 3oz of butter while Betty’s calls for 4 oz. of butter – write the ratio of the
amount of butter in Kim’s recipe to the amount of butter in Betty’s recipe.
ex. Six out of 30 students failed the exam – write a ratio of the students that passed the exam to the
students that took the exam
ex. This last exam 20 students made an A or a B. The rest made below a B.
Write a ratio of the students that made an A or a B to the total number of students.
ex. The cost of building a walkway was $200 for labor and $400 for material. Find the ratio of the
cost of material to the cost of labor.
ex
A baseball team won 120 games and lost 40 games. Write a ratio of the games won to the total
number of games played.
rate: comparison of two quantities with different units
ex. a car is driven for 210 miles on 15 gallons . Write the rate of of miles to gallons ( miles per gallon)
ex. You earn $42 for working 7 hours. Write a rate of the amount you earn per hour.
ex. If the rate of boys to girls in the class room is 2 to 3, then a classroom with 12 boys consists of
how many girls. ?
ex. the rate of accidents in a workplace to the number of days is 2 to 11, then in a period of 121
days you would have how many accidents
113
Unit Rate: is when the denominator is equal to 1.
ex. It costs $3.36 per 15 oz. Write the rate as a unit rate.
ex. A man paid $45 for 3 shirts. Write as a unit rate
ex. A car is driven 240 miles on 12 gallons. Write as a unit rate
ex. You earn $300 for working 60 hours. Write as a unit rate .
ex. A recipe asks for 5 tablespoons of sugar to 2 cups of flour. Write as a rate.
US System of measurement:
See page 431 for more common measurements
Length, mass, volume
1 ft = _________ in. ==> 48 in = _____________ ft.
1 lb. = ___________ oz.
1 cup = _________ oz.
5 lbs = ____________ oz.
1pint = 2 cups ==>
3 pints = _________ oz
Area: units of are ( surface of a region ) : 1ft2 = ____________ in2
1 acre = 43560 ft2
1 mi2 = 640 acres
Dimensional Analysis: converting from one unit to another.
Find the number of gallons of water in a fish tank that is 36in. long , 24 in. wide and is felled to a depth of 16 in. Use the fact
that 1gallon = 231 in3
114
More examples:
1. convert:
2 ½ cups = __________ oz.
14 ft = ____________ yards
1 ½ miles = _________ yards
36 ft2 = _________ yards2
3 ft2 = _____________ in2
30 lbs = __________ oz
1 day = ______________ seconds
2. Use the conversions on page 435 to answer the following questions.
100 yards = _________ meters
If you weigh 140 lbs, then how many kilograms is that? _________
If you are traveling at 100 km/h does that break the 60 mph speed limit ? ____________
Gasoline costs 35.8 cents per liter. How much is that per gallon ?
Proportion: the equality of two ratios or rates
We write a/b and c/d are equal ratios, then a/b = c/d
ex. 2/5 = 14 / 35
1st, 2nd, 3rd, 4th terms:
first and fourth are called the extremes, 2nd and 3rd are called the means
Note: in any proportion, the product of the means = the product of the extremes
115
ex. Solve for x.
x/2 = (x + 1) / 3 → _____________
Solve for n.
Solve for r.
3/n = 25/13 → __________
( r – 2 ) /5 =
(r + 3 ) / 2
More Examples:
1. solve for t.
(t2 + 5) / (3t + 1 ) = t / 3
2. A $2500 investment earns $225 in interest. What investment would earn $350 over the same time
period ?
3. If a person loses 8 lbs in 6 months, then how long would it take him to lose 20 lbs.
4. The dosage for a medicine is 2 mg. for every 80 lbs of body weight. How many milligrams of this
medication are required for a person that weighs 220 lbs ?
5. It takes 12 gallons to drive 200 miles, how many gallons will take to complete a 500 mile trip ?
6. A steak costs $12.60 for 3 lbs. At this rate, how much does an 8 lb. steak cost ?
116
Direct and Inverse Variation.
Direct Variation
: the more you study, the higher your grade
: the greater the distance your drive, the greater the amount of gas you use
y = kx describes a direct relationship between x and y. k is a constant of variation
(constant of proportionality) . We say y varies directly with (as) x.
ex. Find the constant of variation if y varies directly as x and y = 10 when x = 25.
ex. Given that R varies directly as T, and R = 20 when T = 15, find R, when T = 45.
Notice that
when y1 = k1 • x1 and y2 = k2 • x2
if k1 = k2, then
y1/x1 = y2 / x2 .
R = k x Î R varies _______________ as x
If R = 12 when x is 2, then find R when x = 5 . Î __________________
ex. The number of words typed is directly proportional (varies directly) to the time spent typing. A typist
can type 260 words in 4 minutes. Find the number of words typed in 15 minutes.
ex. The distance on object falls is directly proportional to the square of the time (t) of the fall. If an
object falls a distance of 8 ft. in 0.5 second, how far will the object fall in 5 seconds ?
117
Inverse Variation ( Varies inversely – inversely proportional )
: The faster you run, the less time it takes to cover one lap
: The more students in class, the less attention a student gets
y = k/x
describes an inverse relationship between x and y. k is a constant of variation
(constant of proportionality) . We say y varies inversely with (as) x.
R = k / x Î R varies ____________ as x
If R = 4 when x = 1/2, then find R when x = 5/2
Notice that if y1 = k1 /x1 and y2 = k2 / x2 and k1 = k2, then y1 •x1 = y2 •x2
ex. If y varies directly as x2 and y = 12 when x = 2, then find y when x = 3
ex. Let g vary inversely as d and g = 200 when d = 40, then find g when d = 10
ex. The time (t) of travel of an automobile trip varies inversely as the speed (v). Traveling at an
average speed of 65 mph, a trip took 4 hours. The return trip took 5 hours. Find the average speed
of the return trip.
118