MATH085 In-Class Assignment
Contents
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
9.3 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
10.3a Quadratic Formula
10.5 Graphs of Quadratic Equations
6.1 Adding and subtracting polynomials
Group 1: 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 cold’s 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?
Group 2 & 3: (p.348, #103) Use the graph and the polynomials on page 348.
Group 2 use the second degree polynomials and Group 3 use the third degree polynomials to answer the following questions:
- Use the equations to find a model for M – W.
- According to the model you found in part (a), what is the difference in median annual income between men and women with 14 years of education?
- According to the data displayed by the graph on page 348, what is the actual difference between men and women with 14 years of education? Did the model in part (b) over or under estimate this difference? By how much?
Compare your results with the other group. Which model do you think is a better fit for the actual data?
6.2 Multiply polynomials
Groups 1 & 2 (p.359 #103 & 104) (a) Express the area of the large rectangle as the product of the two binomials. (b) Find the sum of the areas of the four smaller rectangles. (c) Use polynomial multiplication to show that your expressions for area in parts (a) and (b) are equal.
Group 1/ Group 2
Group 3 (p.359, #121) Find a polynomial that represents the area of the shaded region.
6.3Special Products
Group 1: (p.366 #97 & 98) The square garden shown on page 366 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.
- Draw a picture of the original garden with the expanded sections. Label the length and width of the expanded garden.
- Write a product of two binomials that expresses the area of the larger garden (with units labeled).
- Write a polynomial in standard form that expresses the area of the larger garden.
Group 2: (p.367 #101 & 102) The square painting in the figure on page 366 measures x inches on each side. The painting is uniformly surrounded by a 1-inch wide frame.
- Write a polynomial in standard form that expresses the area of the square that includes the painting and the frame.
- Write an algebraic expression that describes the area of the frame.
Group 3: (p.367 #118) Express the area of the plane figure shown as a polynomial in standard form.
6.4 Polynomials in several variables
Group 1 (p.374 #88) The storage shed shown on p.374 has a volume given by . A small business is considering the shed installed. The shed’s height, 2x, is 26 feet and its length, y, 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 #89-91) An object that is falling or vertically projected into the air has its height, in feet, above the ground given by where h is the height, in feet, vo is the original velocity of the object, in feet per second, t is the time the object is in motion, in seconds, and ho is the height, in feet, from which the object is dropped or projected. The figure, on p. 374, shows that a ball is thrown straight up from a rooftop at an original velocity of 80 feet/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?
Group 3 (p.375 #109) Find a simplified polynomial in two variables that describes the area of the shaded region.
6.5 Dividing Polynomials
Group 1 (p.384 #87) Polynomial models for the U.S. film box office receipts in millions of dollars,, and admissions in millions of tickets sold,. (a) Use the data from the bar graphs on the bottom of p. 383 to determine the average admission price for a film ticket in the year 2000. (b) Use the models to write an algebraic formula that represents the average admission fee for a film ticket x years after 1980. (c) Use the formula from part (b) to determine the average ticket price for the year 2000. Does it over or under estimate the actual you determined in part (a)?
Group 2 (p.383 #83) Divide the sum of and by.
Group 3 (p.383 # 82) Simplify the expression
6.6 Dividing Polynomials by Binomials
Groups 1 (p.392 #47) 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.
(a) Use the given quotient of polynomials to represent your total salary 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: (p.391#35) Divide as indicated the check your answer by showing that the product of the divisor and quotient , plus the remainder, is the dividend:
Group 3 (p.372 #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
Group #1 (p.405 #149) In 2007, the U.S. population was approximately 3.1 x 108 and each person spent about $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.404 #147) (a) In 2005, the U.S. government collected$2.27 trillion in taxes. Express this number in scientific notation. (b) In 2005, the U.S. population was about 298 million. Express this in scientific notation. (c) Using parts (a) and (b), if this was divided evenly among all Americans, how much would each citizen pay? (Make sure to use the appropriate number of significant digits.)
Group #3 (p.405 #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.)
Chapter 7
7.1 Greatest Common Factor & Factoring by Grouping
GROUP 1: (p.420 #66) Factor by Grouping, show your steps
Group 2: (p.420 #88) Write a polynomial that represents the shaded area in the figure shown on page 420 #88. Then factor the polynomial. The square is 4x on each side.
Group 3: (p.420 #89) 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
Groups 1 & 2: (a) Factor the polynomial completely. (b) Evaluate both the original and its factored form for t = 2. Do you get the same answer? Describe what the answer means in the context of the problem.
Group 1 (p.428 #77) You dive directly upward from a board that is 32 feet high. After t seconds, your height above the water is described by: . (c) What does each of the terms in the original equation represent in the context of the problem? (The book does not talk about this—make an educated guess)
Group 2 (p.429 #78) You dive directly upward from a board that is 48 feet high. After t seconds, your height above the water is described by:. (c) Use the equation to determine the highest point you reached in your dive. (The book does not talk about this—sketch what you think the path of your dive looks like, and then explain how you found the highest point.)
Group 3 (p.429 #96) 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 on page 429 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.436 # 88) Factor completely
Group 2 (p.436 #84) Factor completely
Group 3 (p.436 #92) (a) Factor (b) Use the factorization method you used from part (a) to help you to factor and then simplify each factor.
7.4 Factoring special forms
Find the formula for the area of the shaded region and express it in factored form.
GROUP 1 (p.445 # 102) GROUP 2 (p.445 #100)
GROUP 3 (p.445 #96) Factor completely
7.5 A General Factoring Strategy
GROUP 1 (p.453 #109) (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 π. Then factor this expression completely.
GROUP 2 (p.453 #108) A building has a height represented by xfeet. The building’s base is a square and its volume is cubic feet. Express the building’s dimensions in terms of x.
GROUP 3 (p.453 #107) A rock is dropped from the top of a 256-foot cliff. The height, in feet, of the rock above the water after t seconds is modeled by the polynomial. Factor this expression completely.
7.6 Solving Quadratic Equations by Factoring
GROUP 1 (p.463 #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.464 #72&73) The formula models spending by international travelers to the U.S., S, in billions of dollars, x years after 2000. (a) In which years did international travelers spend $72 billion? (b) In which years did international travelers spend $66 billion?
GROUP 3 (p.465 #86) A vacant rectangular lot is being turned into a community vegetable garden measuring 15 meters by 12 meters. A path of uniform width is to surround the garden. If the area of the lot is 378 square meters, find the width of the path surrounding the garden. (Draw a diagram to help you solve this problem.)
Chapter 8
8.1 Rational expressions and their simplification
GROUP 1 (p.482 #86) The rational expression describes the cost, indollars, to removex 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, 50, and 80. Describe the meaning of each evaluation in terms of percentage of pollutants removed and cost. (b) For what value of xis the expression undefined? (c) What happens to the cost as xapproaches 100%? How can you interpret this observation?
GROUP 2 (p.482 #90) A company that manufactures small canoes has costs given by the equation in which x is the number of canoes made and C is the cost to make each canoe. (a) Find the cost per canoe when making 100 canoes. (b) Find the cost per canoe when making 10,000 canoes. (c) Does the cost per canoe increase or decrease as more canoes are made? Explain why this happens.
GROUP 3 (p.483 #93) The mathematical model for calories needed per day, W, by women in age group x to maintain a sedentary lifestyle is . The model for calories needed per day, M, by men in age group x to maintain a sedentary lifestyle is . (a) Determine the calories needed for women and for men between the ages of 19 and 30 (age group 4). (b) Write a simplified expression for the ratio between the calories needed by women as compared with men. (c) Write a simplified expression for the difference in calories needed by women as compared to men.
8.2 Multiplying and Dividing Rational Expressions
GROUPs 1-3 Write a simplified rational expression for the area of the given figures. Assume all measures are in inches.
GROUP 1 (p.490 #73) Rectangle with width of and length of
GROUP 2 (p.490 #74) Rectangle with width of and length of
GROUP 3 (p.490 # 75) Triangle with height of and base of
8.3 Adding and Subtracting Rational Expressions with Same Denominator
GROUP 1 (p.498 #73) Anthropologists and forensic scientists classify skulls using where L is the skull’s 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.498 #76) Find the perimeter of the rectangle, which is measured in inches.
GROUP 3 (p.498 #74) The temperature, in degrees Fahrenheit, of a dessert placed in a freezer for thours 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.508 #102) Express the perimeter as a single rational expression.
Groups 2 & 3: The two formulas: Young’s Rule and Cowling’s Rule approximate dosage of a drug prescribed to children. In each formula, A is the child’s age in years, D is an adult dosage and C is the proper child’s dosage. The formulas apply for ages 2 through 13, inclusive.
GROUP 2 (p.508 #94) Use Young’s Rule to find a child’s 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 Use Cowling’s Rule to find the difference in a child’s 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.
Chapter 9
9.1 Finding Roots
GROUP 1 (p.556 #86) The formula models the maximum safe speed, v in mph, 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 a highway exit ramp as 360 feet. What is the maximum safe speed?
GROUP 2 (p.556 #87) 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 car’s skid marks to be 245 feet long. Estimate the speed at which the motorist was traveling before braking. If the posted speed is 50 mph and the motorist said that she was not speeding, should the officer believe her? Explain.
GROUP 3 (p.556 #89) Please read what is said on page 556 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 556 to determine whether the model describes healthy children or those with severe autistics.
9.2 Multiplying and Dividing Radicals
GROUP 1 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 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.566 #118) Express the area of the rectangle that has a length of feet and width of feet as a square root in simplified form.
ALL GROUPS Simplify:
(p.565 #83) (p.565 #96) (p.566 #105)