Idea Transcript
MAT085 In-Class Assignment 6.1 Adding and subtracting polynomials 6.2 Multiply polynomials 6.3 Special Products 6.4 Polynomials in several variables 6.5 Dividing Polynomials 6.6 Dividing Polynomials by Binomials 6.7 Negative exponents and scientific notations 7.1 Greatest common factor & factoring by grouping 7.2 Factoring trinomials whose leading coefficient is one 7.3 Factoring trinomials whose leading coefficient is not one 7.4 Factoring special forms 7.5 A general factoring strategy 7.6 Solving quadratic equations by factoring 8.1 Rational expressions and their simplification 8.2 Multiplying and dividing rational expressions 8.3 Adding and subtracting rational expressions with same denominator 8.4 Adding and subtracting rational expressions with different denominators 9.1 Finding roots 9.2 Multiplying and dividing radicals 9.3a Operations with radicals 9.4 Rationalize the denominator 9.5 Radical equations 9.6 Rational exponents 10.1a Solving quadratic equations using square root property 10.1b Solving quadratic equations using square root property 10.2 Solving quadratic equations by completing the square 10.3 Quadratic formula 10.4 Imaginary numbers 10.5 Graphs of quadratic equations
6.1 Adding and subtracting polynomials
(p.329, #97) The common cold is caused by a rhinovirus. The polynomial describes the billions of viral particles in our bodies after days of invasion. Find the number of viral particles, in billions, after 0 days (the time of the colds onset when we are still feeling well), 1 day, 2 days, 3 days, and 4 days. After how many days is the number of viral particles at a maximum and consequently we feel the sickest? By when should we feel completely better?
(p.329, #101) Use the graph on page 329, second column to answer the following questions. a. Use the graph to estimate, to the nearest percent, womens earnings as a percentage of mens in 2000. b. Use the mathematical model: to find womens earnings as a percentage of mens in 2000. c. In 2000, median annual earnings for US women and men were $27,355 and $37,339, respectively. What were womens earnings as a percentage of mens? Use a calculator and round to the nearest tenth of a percent. How well do your answers in part (a) and (b) model the actual data? 6.2 Multiply polynomials
1. (p.340 #103) (a) Express the area of the large rectangle as the product of the two binomials. (b) Find the sum of the areas of each of the four small rectangles. (c) Use polynomial multiplication to show that your expressions for area in parts (a) and (b) are equal.
2. (p.340 #104) (a) Express the area of the large rectangle as the product of the two binomials. (b) Find the sum of the areas of each of the four small rectangles. (c) Use polynomial multiplication to show that your expressions for area in parts (a) and (b) are equal.
3. (p.341, #115) Find a polynomial that represents the area of the shaded region.
6.3 Special Products (1) Provide an example for multiplying the sum and difference of two terms using FOIL. (2) Provide an example of the short-cut for finding the square of a binomial sum, and the square of a binomial difference.
1. (p.348 #97 & 98) The square garden shown on page 348 measures x yards on each side. The garden is to be expanded so that one side is increased by 2 yards and an adjacent side is increased by one yard. a. Draw a picture of the original garden with the expanded sections. Label the length and width of the expanded garden. b. Write a product of two binomials that expresses the area of the expanded garden (with units labeled). c. Write a polynomial in descending powers of x that expresses the area of the expanded garden. 2. (p.349 #101 & 102) The square painting in the figure on page 349 measures x inches on each side. The painting is uniformly surrounded by a frame that measures 1 inch wide. a. Write a polynomial in descending powers of x that expresses the area of the square that includes the painting and the frame. b. Write an algebraic expression that describes the area of the frame. 3. (p.349 #111) Express the area of the plane figure shown as a polynomial in standard form. x
6.4 Polynomials in several variables (1) What does it mean to evaluate a polynomial in several variables? Provide an example. (2) Describe how to determine the degree of a polynomial that has more than one variable, and provide an example. (3) Show how to add / subtract polynomials with more than one variable. (3) Show how to multiply polynomials with two variables.
Group 1 (p.356 #88) The storage shed shown on page 356 has a volume given by . A small business is considering having a shed installed like the one shown on page 356. The sheds height , , is 26 feet and its length, , is 27 feet. Find the volume of the storage shed. If the business requires at least 18,000 cu. ft. of storage space, should they construct this shed?
Group 2 (p.356 #87) The number of board feet, , that can be manufactured from a tree with a diameter of inches and length of feet is modeled by the formula . A building contractor estimates that 3000 board feet of lumber is needed for a job. The lumber company has just milled a fresh load of timber from 20 trees that averaged 10 inches in diameter and 16 feet in length. Is this enough to complete the job? If not, how many additional board feet of lumber are needed?
Group 3 (p.356 #89-91) An object that is falling or vertically projected into the air has its height, in feet, above the ground given by where is the height, in feet, is the original velocity of the object, in feet per second, is the time the object is in motion, in seconds, and is the height, in feet, from which the object is dropped or projected. The figure, on page 356, shows that a ball is thrown straight up from a roof top at an original velocity of 80 feet per second from a height of 96 feet. The ball misses the rooftop on its way down and eventually strikes the ground. How high above the ground will the ball be 2, 4 and 6 seconds after being thrown? 6.5 Dividing Polynomials
Group 1 (p.365 # 82) Simplify the expression
Group 2 (p.365 #83) Divide the sum of
and
by
.
Group 3 (p.365 #85) Simplify the expression 6.6 Dividing Polynomials by Binomials
Groups 1 & 2 (p.373 #47 & 48) You just signed a contract for a new job. The salary for the first year is $30,000 and there is to be a percent increase in your salary each year. The expression describes your salary over n years, where x is the sum of 1 and the yearly percent increase, expressed as a decimal.
Group 1: (a) Write the polynomial that represents your income over three years, and then (b) simplify that expression by performing the division. (c) Suppose you are to receive an increase of 5% per year. (x = 1 + 0.05). Substitute 1.05 into the expression for x in both expressions, from part (a) and part (b). What is your total salary over the three years?
Group 2: (a) Write the polynomial that represents your income over four years, and then (b) simplify that expression by performing the division. (c) Suppose you are to receive an increase of 8% per year. (x = 1 + 0.08). Substitute 1.08 into the expression for x in both expressions, from part (a) and part (b). What is your total salary over the three years? Group 3 (p.373 #45) Draw a picture to help you solve this problem. Write a simplified polynomial that represents the length of the rectangle when its area is square inches and its width is
inches.
6.7 Negative exponents and scientific notations Please let someone from your group who has not put a problem on the board yet put the problem up for your group.
Group #1 (p.386 #149) If the U.S. population is 2.9 x 108 and each person spends abut $120 per year on ice cream, express the total annual spending on ice cream in scientific notation. (Make sure to use the appropriate number of significant digits.)
Group #2 (p.386 #150) A human brain contains 3 x 1010 neurons and a gorilla brain contains 7.5 x 109 neurons. How many times as many neurons are in the brain of the human as compared with the brain of the gorilla? (Make sure to use the appropriate number of significant digits.)
Group #3 (p.386 #151) Use the motion formula d = rt, distance equals rate times time, and the fact that light travels at the rate of 1.86 x 105 miles per second. If the moon is approximately 2.325 x 105 miles from Earth, how many seconds does it take moonlight to reach Earth? (Make sure to use the appropriate number of significant digits.)
Everyone: (p.386 #152) Use the motion formula d = rt, distance equals rate times time, and the fact that light travels at the rate of 1.86 x 105 miles per second. If the sun is approximately 9.14 x 107 miles form Earth, how many seconds, to the nearest tenth of a second, does it take sunlight to reach Earth? (Make sure to use the appropriate number of significant digits.) 7.1 Greatest common factor & factoring by grouping (1) Describe how factoring a polynomial is like undoing the distributive property, and provide an example. (2) Describe what is meant by Factoring by grouping, and provide an example.
GROUP 1: (p.402 # 72) Factor by Grouping, show your steps Group 2: (p.402 #94) Write a polynomial that represents the shaded area in the figure. Then factor the polynomial. The square is 4x on each side. Group 3: (p.402 #95) An explosion causes debris to rise vertically with an initial velocity of 64 feet per second. The polynomial describes the height of the debris above the ground, in feet, after x seconds. (a) Find the height of the debris after 3 seconds. (b) Factor the polynomial. (c) Use the factored form of the polynomial to find the height after 3 seconds. Do you get the same answer as you did for part (a)? If so, does this prove that your factorization is correct? Explain. 7.2 Factoring trinomials whose leading coefficient is one (1) Explain how to factor a trinomial with a leading coefficient of one. (2) Describe the steps you need to consider in order to factor a trinomial completely.
Group 1 (p.410 #77) You dive directly upward from a board that is 32 feet high. After t seconds, your height above the water is described by: . (a) Factor the polynomial completely; begin by factoring out the GCF. (b) Evaluate both the original polynomial and its factored form for t = 2. Do you get the same answer? Describe what the answer means in the context of the problem. (c) What does each of the terms in the original equation represent in the context of the problem? (The book does not talk about thismake an educated guess)
Group 2 (p.410 #78) You dive directly upward from a board that is 48 feet high. After t seconds, your height above the water is described by: . (a) Factor the polynomial completely; begin by factoring out the GCF. (b) Use the equation to determine the highest point you reached in your dive. (The book does not talk about thissketch what you think the path of your dive looks like, and then explain how you found the highest point.)
Group 3 (p.411 #88) A box with no top is to be made from an 8-inch by 6-inch piece of metal by cutting identical squares from each corner and turning up the sides. The volume of the box is modeled by: . Factor the polynomial completely. Then use the dimensions given on the box below and show that its volume is equivalent to the factorization that you obtained.
7.3 Factoring trinomials whose leading coefficient is not one
Group 1 (p.418 # 88) Factor completely
Group 2 (p.417 #84) Factor completely (Each table in the group should choose a different factor method and display those methods on the board.)
Group 3 (p.418 #92) (a) Factor
(b) Use the factorization method you
used from part (a) to help you to factor each factor.
and then simplify
7.4 Factoring special forms
Find the formula for the area of the shaded region and express it in factored form. GROUP 1 (p.425 # 100) GROUP 2 (p.425 #102)
GROUP 3 (p.425 #96) Factor completely 7.5 A general factoring strategy
GROUP 1 (p.434 #105) (The arrows originate at the center of each of these concentric circles.) Express the area of the shaded ring shown in the figure in terms of . Then factor this expression completely.
GROUP 2 (p.434 #104) A building has a height represented by x feet. The buildings base is a square and its volume is Express the buildings dimensions in terms of x.
cubic feet.
GROUP 3 (p.434 #103) A rock is dropped from the top of a 256foot cliff. The height, in feet, of the rock above the water after t seconds is modeled by the polynomial completely
. Factor this expression
7.6 Solving quadratic equations by factoring Describe the zero-product principle and why it is helpful when solving a quadratic equation.
GROUP 1 (p.444 #70-71) An explosion causes debris to rise vertically with an initial velocity of 72 feet per second. The formula describes the height of the debris above the ground, h feet, t seconds after the explosion. (a) How long will it take for the debris to hit the ground? (b) When will the debris be 32 feet above the ground?
GROUP 2 (p.444 #72-73) The formula models the number of inmates, N, in thousands, in the US state and federal prisons x years after 1980. The graph of the formula is shown on page 444. (a) In which year were there 740 thousand inmates in the US state and federal prisons? (b) In which year were there 1100 thousand inmates in US state and federal prisons?
GROUP 3 (p.444 #78-79) The formula describes the number of football games, N, that must be played in a league with t teams if each team is to play every other team once. (a) If the league has 36 games scheduled, how many teams belong to the league, assuming that each team plays every other team once? (b) If the league has 45 games scheduled, how many teams belong to the league, assuming that each team plays every other team once? 8.1 Rational expressions and their simplification
GROUP 1 (p.462 #86) The rational expression describes the cost, in dollars, to remove x percent of the air pollutants in the smokestack emission of a utility company that burns coal to generate electricity. (a) Evaluate the expression for x = 20, x =50, and x = 80. Describe the meaning of each evaluation in terms of percentage of pollutants removed and cost. (b) For what value of x is the expression undefined? (c) What happens to the cost as x approaches 100%? How can you interpret this observation?
GROUP 2 (p.462 #90) A company that manufactures small canoes has costs given by the equation in which x is the number of canoes manufactured and C is the cost to manufacture each canoe. (a) Find the cost per canoe when manufacturing 100 canoes. (b) Find the cost per canoe when manufacturing 10,000 canoes. (c) Does the cost per canoe increase or decrease as more canoes are manufactured? Explain why this happens.
GROUP 3 (p.462&3 #91&92) A drug is injected into a patient and the concentration of the drug in the blood stream is monitored. The drugs concentration, y, in milligrams per liter, after x hours is modeled by . The graph of this equation is on the top left corner of page 463. (a) Use the equation to find the drugs concentration after 3 hours. Then identify the point on the equations graph that conveys this information. (b) Use the graph of the equation to find after how many hours the drug reaches its maximum concentration. Then use the equation to find the drugs concentration at this time. 8.2 Multiplying and dividing rational expressions (1) Describe how to multiplying rational expressions and multiplying fractions are the same/different, and provide an example. (2) Explain how dividing rational expressions and dividing fractions are the same/different and provide an example,
GROUP 1 (p.469 # 62) GROUP 2 (p.469 #64) GROUP 3 (p.469 # 73, 74) In section 8.1 there were two problems which provided formulas. The first one, describes the cost, in millions of dollars, to remove x percent of pollutants that are discharged into the river. The second one is the cost in dollars to manufacture each of x bicycles. In both cases, we were wrong. The cost will be half of what we originally anticipated. Write a rational expression that represents the reduced cost for each of these problems. 8.3 Adding and subtracting rational expressions with same denominator
GROUP 1 (p.477 #73) Anthropologists and forensic scientists classify skulls using where L is the skulls length and W is its width. (a) Express the classification as a single rational expression. (2) If the value of the rational expression in part (a) is less than 75, a skull is classified as long. A medium skull has a value between 75 and 80, and a round skull has a value of over 80. Use your rational expression from part (a) to classify a skull that is 5 inches wide and 6 inches long. GROUP 2 (p.477 #76) Find the perimeter of the rectangle. GROUP 3 (p.477 #74) The temperature, in degrees Fahrenheit, of a dessert placed in a freezer for t hours is modeled by . (a) Express the temperature as a single rational expression. (b) Use your rational expression from part (a) to find the temperature of the dessert, to the nearest hundredth of a degree, after 1 hour and after 2 hours. 8.4 Adding and subtracting rational expressions with different denominators
GROUP 1 (p.488 #102) Express the perimeter as a single rational expression. Groups 2 & 3: The two formulas: Youngs Rule and Cowlings Rule approximate dosage of a drug prescribed to children. In each formula, A is the childs age in years, D is an adult dosage and C is the proper childs dosage. The formulas apply for ages 2 through 13, inclusive.
GROUP 2 (p.488 #94) Use Youngs Rule to find a childs dosage for a 10-year old child and a 3-year old child. Find the difference in these dosages and express the answer in a single rational expression in terms of D. Then describe what your answer means in terms of the variables in the model.
GROUP 3 (p.488 #96) Use Cowlings Rule to find the difference in a childs dosage for a 10-year old child and a 3-year old child. Find the difference in these dosages and express the answer in a single rational expression in terms of D. Then describe what your answer means in terms of the variables in the model.
ALL Groups Compare Groups 2s and Group 3s answers. Discuss the differences and describe other factors that might be used when determining a childs dosage. Is this factor more or less important than age? Explain why. 9.1 Finding roots (1) What is meant by a square root, explain and provide an example. (2) Describe the Principle nth root of a real number and provide a couple of examples. {If a is a nonnegative real number, the nonnegative number b such that
, denoted by
, is the principal square root of a.
GROUP 1 (p.538 #86) The formula models the maximum safe speed, v, in miles per hour, at which a car can travel on a curved road with radius of curvature r, in feet. A highway crew measures the radius of curvature at an exit ramp on a highway as 360 feet. What is the maximum safe speed? GROUP 2 (p.538 #88) Police use the formula to estimate the speed of a car, v, in miles per hour, based on the length, L, in feet of its skid marks upon sudden braking on a dry asphalt road. A motorist is involved in an accident. A police officer measures the cars skid marks to be 45 feet long. Estimate the speed at which the motorist was traveling before braking. If the posted speed is 35 mph and the motorist tells the officer that she was not speeding, should the officer believe her? Explain.
GROUP 3 (p.538 #89) Please look in your book and read what is said on page 538 concerning the line graph. The data for one of the two groups shown by the graphs can be modeled by where y is the head circumference, in centimeters, at age x months, . (a) According to the model, what is the head circumference, in centimeters at birth? (b) According to the model, what is the head circumference at 9 months? (c) According to the model, what is the head circumference at 14 months? Use a calculator to round the nearest tenth of a centimeter. (d) Use the values in parts (a) through (c) and the graphs on page 538 to determine whether the model describes healthy children or those with severe autistics. 9.2 Multiplying and dividing radicals (1) Describe the Product Rule for Square Roots, and provide an example. (2) Describe how to simplify square roots with variables to even powers, and provide an example. (3) Describe how to simplify square roots with variables to odd powers. (4) Describe the Quotient Rule for Square Roots, and provide an example. (5) Describe the Product and Quotient Rules for nth roots and provide an example.
GROUP 1 (p.547 #117) The algebraic expression is used to estimate the speed of a car, in miles per hour, prior to an accident based on the length of its skid marks L, in feet. Find the speed of a car that left skid marks 40 feet long and write the answer in simplified radical form. GROUP 2 (p.547 #118) the time, in seconds, that it takes an object to fall a distance d, in feet, is given by the algebraic expression . Find how long it will take a ball dropped from the top of a building 320 feet tall to hit the ground. Write the answer in simplified radical form. GROUP 3 (P.547 #119) Express the area of the rectangle that has a length of feet and width of as a square root in simplified form. ALL GROUPS Simplify: (p.546 #83)
(p.546 #96) (p.547 #105)
9.3a Operations with radicals (1) Explain what is meant by like radicals and give an example.
(2) Explain how to add/subtract like radicals and provide examples.
All groups: find the area and the perimeter of the polygon. GROUP 1 GROUP 2
GROUP 3
9.3b Operations with radicals
All groups: Find the area and the perimeter of the polygon. GROUP 1 GROUP 2
GROUP 3
9.4 Rationalize the denominator Describe what is meant by rationalizing the denominator, and provide an example.
GROUP 1 (p.559, #83) Do you expect to pay more taxes than were withheld? Would you be surprised to know that the percentage of taxpayers who receive a refund and the percentage of taxpayers who pay more taxes vary according to age? The formula models the percentage, P, of taxpayers who are x years old who must pay more taxes. (a) What percentage of 25-year olds must pay more taxes? (b) Rewrite the formula by rationalizing the denominator. (c) Use the rationalized form of the formula to find the percentage of 25-year olds who must pay more taxes. Do you get the same answer? If so, does this prove that you correctly rationalized the denominator? Explain. GROUP 2 (p.559, #85) The early Greeks believe that the most pleasing of all rectangles were the golden rectangles, whose ratio of width to height is . Rationalize the denominator for this ratio and then use a calculator to approximate the answer to the nearest hundredth. GROUP 3 (p.560, #92) Simplify 9.5 Radical equations (1) Describe in your own words, the steps used to solve a radical equation, provide an example.
GROUP 1 (p.566 #58) The time, t, in seconds for a free-falling object to fall d feet is modeled by the formula . If a worker accidentally drops a hammer from a building and it hits the ground after 4 seconds, from what height was the hammer dropped? GROUP 2 (p.567 #60) The pendulum on a grandfather clock has a length of l feet. The time, t, in seconds, it takes the pendulum of the clock to swing through one complete cycle is described by . Determine how long the pendulum must be for one complete cycle to take 2 seconds. Round your answer to the nearest hundredth of a foot. GROUP 3 (p.567 #61) Two tractors are removing a tree stump from the ground. If two forces, A and B, pull at right angles to each other, the size of the resulting force, R, is given by the formula: . Tractor A exerts 300 pounds of force. If the resulting force is 500 pounds, how much force is tractor B exerting in the removal of the stump? 9.6 Rational exponents (1) What is a rational exponent? How is it the same/different than a regular exponent? Provide examples.
GROUP 1 (p.572 # 58) The formula models the wind speed, v, in miles per hour, needed to produce p watts of power from a windmill. How fast must the wind be blowing to produce 120 watts of power? GROUP 2 (p.572 # 60) According to the AMA, the percentage of potential employees testing positive for illegal drugs in on the decline. The formula models the percentage, P, of people applying for jobs who tested positive t years after 1985. (a) What percentage of people applying for jobs tested positive for illegal drugs in 1985? (b) Using this formula, how many in 2007?
GROUP 3 (p.572 # 44) Simplify
by first converting it to radical form.
10.1a Solving quadratic equations using square root property
GROUP 1 (p.588 #79) A square flower bed is to be enlarged by 2 meters on each side. If the larger square has an area of 144 square meters, what is the length of the original square?
GROUP 2 (p.588 #81) A machine produces open boxes using square sheets of metal. The figure, on the bottom left of page 588, illustrates that the machine cuts equalsized squares measuring 2 inches on a side from the corners and then shapes the metal into an open box by turning up the sides. If each box must have a volume of 200 cubic inches, find the size of the length and width of an open box.
GROUP 3 (p.587 #74) If the area of a circle is
square inches, find its radius.
(You should have this memorized: the formula for area of a circle is
).
10.1b Solving quadratic equations using square root property
GROUP 1 (p.587 #67) A ladder is leaning up against the side of building. The bottom of the ladder is 10 feet from the base of the building, and the top of the ladder reaches a height of 8 feet. How long is the ladder? GROUP 2 (p.587 #69) A baseball diamond is actually a square with 90-foot sides. What is the distance between home plate and third base? GROUP 3 (p.587 #72) In a 27-inch square television set, the length of the screens diagonal is 27 inches. Find the measure of the sides of the screen. Express your answer in exact form, and then approximate to the nearest inch. Everyone, after working on your group problem try this one: A small skateboard companys revenue is modeled by where P is the companys daily profit and q is the quantity (number) of skateboards produced each day. Solve for q when P = 136. Explain your answer in the context of the problem. 10.2 Solving quadratic equations by completing the square
GROUP 1 (p.593 #10) Complete the square for the binomial resulting perfect square trinomial.
, then factor the
GROUP 2 (p.593 #30) Solve the quadratic equation completing the square.
by
GROUP 3 (p.593 #34) Solve the quadratic equation completing the square.
by
10.3 Quadratic formula
GROUP 1: (p.603 #71) A rectangular vegetable garden is 5 feet wide and 9 feet long. The garden is to be surrounded by a tile border of uniform width. If there are 40 square feet of tile for the border, how wide, to the nearest tenth of a foot, should it be? (Draw a diagram of the garden and the path, labeling the important dimensions that will help you to solve this problem.) GROUP 2: (p.602 #54) A person standing on a platform fires a gun straight up into the air from a height of 50 feet. The formula describes the bullets height above the ground, h, in feet, t seconds after the gun is fired. How long will it take for the bullet to hit the ground? Round your answer to the nearest tenth of a second. What do the numbers in the formula represent in the context of the problem?
GROUP 3: (p.602 #60) The length of a rectangle is 2 meters longer than the width. If the area is 10 square meters, find the rectangles dimensions and the length of the rectangles diagonal (round to the nearest tenth of a meter). Draw the rectangle and label the length, width and diagonal. 10.4 Imaginary numbers
GROUP 1 (p.608 # 44) A football is kicked straight up froma height of 4 feet with an initial speed of 60 feet per second. The formula , describes the balls height above the ground, h, in feet, t seconds after it is kicked. Will the ball reach a height of 80 feet?
GROUP 2 (p.607 #38) Solve the equation GROUP 3 (p.607 #41) Solve the equation 10.5 Graphs of quadratic equations
GROUP 1 (p.617 #51) You have 120 feet of fencing to enclose a rectangular plot that borders on a river. If you do not fence the side along the river, find the length and width of the plot that will maximize the area. What is the largest area that can be enclosed? (make a sketch) GROUP 2 (p.617 #50) A quarterback throws a football to a receiver 40 yards away. The formula models the footballs height y feet above the ground when it is x yards from the quarterback. How many yards from the quarterback does the football reach its greatest height? What is that height? GROUP 3 (p.617 #62) A parabola has x-intercepts at 3 and 7, a y-intercept at -21, and (5, 4) for its vertex. Write the parabolas equation.