* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Vincent's theorem wikipedia, lookup

Recurrence relation wikipedia, lookup

Elementary algebra wikipedia, lookup

Elementary mathematics wikipedia, lookup

Factorization wikipedia, lookup

History of algebra wikipedia, lookup

Signal-flow graph wikipedia, lookup

Partial differential equation wikipedia, lookup

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