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example? Explain why or why not. Developing Skills In 3–12, y varies directly as x. In each case: a. What is the constant of variation? b. Write an equation for y in terms of x. c. Using an appropriate scale, draw the graph of the equation written in part b. d. What is the slope of the line drawn in part c? 3. The per... |
on the test in words per minute. d. Draw a graph that compares the number of words Russ typed to the number of min- utes that he typed. (Let the horizontal axis represent minutes and the vertical axis represent words.) In 15–20, determine if the two variables are directly proportional. If so, write the equation of var... |
(6, 3)G (–3, 1)E 1 –1 –1 O 1 F(0, –2) x Together, the three sets of points form the entire plane. To graph an inequality in the coordinate plane, we proceed as follows: 1. On the plane, represent the plane divider, for example, y 3, by a dashed line to show that this divider does not belong to the graph of the half-pl... |
x as a solid line. Then we shade the region above the line to include the points for which y 2x. y y > 2x y = 2x O x Graph of y > 2x When the equation of a line is written in the form y mx b, the halfplane above the line is the graph of y mx b and the half-plane below the line is the graph of y, mx 1 b. To check whethe... |
–2 O x 2 –2 y 2 O –2 y 2 1 = x x 2 –2 O –2 y 2 O –. x < 1 c. y > 1 d. y < 1 –2 a. x > 1 EXERCISES Writing About Mathematics 1. Brittany said that the graph of 2x y 5 is the region above the line that is the graph of 2x y 5. Do you agree with Brittany? Explain why or why not. 2. Brian said that the union of the graph o... |
The distance to school (y) is at least 2 miles more than the distance to the library (x). 30. The height of the flagpole (y) is at most 4 feet more than the height of the oak tree (x) nearby. 31. At the water park, the cost of a hamburger (x) plus the cost of a can of soda (y) is greater than 5 dollars. 9-10 GRAPHS IN... |
were graphed: y (–4, 6) y = x + 2 (5, 7) (3, 5) (–2, 4) (–1, 3) (1, 3) (0, 2) x Calculator Solution (1) Enter the equation into Y1: (2) Graph to the standard window: ENTER: Y MATH ENTER ENTER: ZOOM 6 X,T,,n ) 2 DISPLAY DISPLAY: EXAMPLE 2 Draw the graph of x y 3. Solution By the definition of absolute value, x x and y ... |
y x c is the graph of y x shifted c units to the right. Reflection Rule for Absolute Value Functions The graph of y x is the graph of y x reflected across the x-axis. Scaling Rules for Absolute Value Functions When c 1, the graph of y cx is the graph of y x stretched vertically by a factor of c. When 0 c 1, the graph ... |
x 3 11. x 2y 7 15. 2x 4y 6 0 In 16–19, describe the translation, reflection, and/or scaling that must be applied to y x to obtain the graph of each given function. 17. y 19. y x 1.5 4 18. y x 2 3 16. y –x 4 22ZxZ 1 2 9-11 GRAPHS INVOLVING EXPONENTIAL FUNCTIONS A piece of paper is one layer thick. If we place another p... |
the curve, we are assuming that the domain of the independent variable, x, is the set of real numbers. You will learn about powers that have exponents that are not integers in more advanced math courses. x 0 1 2 3 2x 20 21 22 23 y 2x 2, 4) (1, 2) 1 (0, 1) O 1 2 3 x (–3, )1 8 (–1, )1 2 (–2, )1 4 –3 –2 –1 There are many... |
Doubling the number of layers in the stack is an increase equal to the size of the stack, that is, an increase of 100%. An exponential change can be a decrease as well as an increase. Consider this example. Start with a large piece of paper. If we cut the paper into two equal parts, each part is one-half of the origin... |
exponential growth. Use the formula A P (1 r)n with r 0.05, P the initial investment, and n 10 years. A 2,000(1 0.05)10 2,000(1.05)10 3,257.78954 Answer The value of the investment is $3,257.79 after 10 years. EXAMPLE 3 The population of a town is decreasing at the rate of 2.5% per year. If the population in the year ... |
.5x, [4, 4] 9. y 5 1.25x, [0, 10] A 10. Compare each of the exponential graphs drawn in exercises 4 through 9. A B B a. What point is common to all of the graphs? b. If the value of the base is greater than 1, in what direction does the graph curve? c. If the value of the base is between 0 and 1, in what direction does... |
if you planned to work 25 days? Explain why or why not. CHAPTER SUMMARY Chapter Summary 393 The solutions of equations or inequalities in two variables are ordered pairs of numbers. The set of points whose coordinates make an equation or inequality true is the graph of that equation or inequality. The graph of a linea... |
and y x can be translated, reflected, or scaled to graph other linear and absolute value functions. Vertical translations (c 0) Horizontal translations (c 0) Reflection across the x-axis Vertical stretching (c 1) Vertical compression (0 c 1) Linear Function Absolute Value y x c (up) y x c (down) y x c (left) y x c (ri... |
3x Review Exercises 395 11. x 2y 8 12. y x 2 13. 2x y 4 In 14–22, refer to the coordinate graph to answer each question. 14. What is the slope of line k? 15. What is the x-intercept of line k? 16. What is the y-intercept of line k? y 1 O 1 k m x 17. What is the slope of a line that is parallel to line k? 18. What is t... |
) (4) (4, 0) (1) The line has no slope. (3) 4 (2) 0 (4) 4 30. In which ordered pair is the x-coordinate 3 more than the y-coordinate? (1) (1, 4) (2) (1, 3) (3) (3, 1) (4) (4, 1) 396 Graphing Linear Functions and Relations 31. Which of the following is not a graph of a function? y 1 O (1) (2) (3) (4 32. Using the domain... |
charge for 1 week for a child who was picked up at the fol- lowing times: Monday at 5:00, Tuesday at 5:10, Wednesday at 6:00, Thursday at 5:00, and Friday at 5:25? 40. Each time Raphael put gasoline into his car, he recorded the number of gallons of gas he needed to fill the tank and the number of miles driven since t... |
equation that states the relationship between the number of chirps per minute of the tree cricket, c, to the temperature in degrees Fahrenheit, F. (To change chirps per minute to chirps in 15 seconds, divide c by 4.) b. Draw a graph of the data given in the table. Record the number of chirps per minute on the horizont... |
4 (3) 4 b (4) 4 b 1 4 3. In decimal notation, 8.72 10–2 is (1) 87,200 (2) 872 (3) 0.0872 (4) 0.00872 4. Which equation is not an example of direct variation? y x 5 2 (1) y 2x y 5 2 x (2) (3) (4) y 5 x 2 5. The graph of y 2x 4 is parallel to the graph of (1) y 2x 5 (2) 2x y 7 (3) y 2x 3 (4) 2x y 0 6. The area of a tria... |
. A square, ABCD, has a vertex at A(4, 2). Side AB is parallel to the x-axis and AB 7. What could be the coordinates of the other three vertices? Explain how you know that for the coordinates you selected ABCD is a square. Part III Answer all questions in this part. Each correct answer will receive 3 credits. Clearly i... |
also want to include design features that harmonize with the rest of the building and its surroundings. Solving a problem such as the design of a ramp often involves writing and determining the solution of several equations or inequalities at the same time. CHAPTER 10 CHAPTER TABLE OF CONTENTS 10-1 Writing an Equation... |
4x 7 Answer y 4x 7 In standard form (Ax By C), the equation of the line is Note: 4x y 7. Writing an Equation Given Slope and One Point 403 EXERCISES Writing About Mathematics 1. Micha says that there is another set of information that can be used to find the equation of a line: the y-intercept and the slope of the lin... |
(2, 1). d. perpendicular to the line 2y 6x 9, and passes through point (2, 1). e. parallel to the line y 4x 3, and has the same y-intercept as the line y 5x 3. f. perpendicular to the line y 4x 3, and has the same y-intercept as the line y 5x 3. 404 Writing and Solving Systems of Linear Functions 11. For a–c, write an... |
draw a line using two points on the line. With this same information, we can also write an equation of a line. Writing an Equation Given Two Points 405 EXAMPLE 1 Write an equation of the line that passes through points (2, 5) and (4, 11). Solution METHOD 1. Use the definition of slope and the coordinates of the two po... |
answer and show that you are correct. 2. Name the principle used in each step of the solution of the equation in Method 2. Developing Skills In 3–6, in each case write an equation of the line, in the form y mx b, that passes through the given points. 3. (0, 5), (2, 0) 4. (0, 3), (1, 1) 5. (1, 4), (3, 8) 6. (3, 1), (9,... |
repair bill at Chickie’s Service Spot includes a fixed charge for an estimate of the repairs plus an hourly fee for labor. Jack paid $123 for a TV repair that required 3 hours of labor. Nina paid $65 for a DVD player repair that required 1 hour of labor. a. Write two ordered pairs (x, y), where x represents the number... |
5 3 5x 2 3 408 Writing and Solving Systems of Linear Functions The slope-intercept form of the equation shows that the y-intercept is 3. Recall from Section 9-6 that we can write this equation in intercept form to show both the x-intercept and the y-intercept. Start with the equation: Add 3x to each side: Divide each ... |
ISES Writing About Mathematics 1. a. Can the equation y 4 be put into the form cept of the line? Explain your answer. y x b 5 1 a 1 b. Can the equation x 4 be put into the form intercept of the line? Explain your answer. y x b 5 1 a 1 to find the x-intercept and y-inter- to find the x-intercept and y- 2. Can the equati... |
sides: y x 1 If we also know that the length of the rectangle is 1 foot more than the width, the dimensions of the rectangle can be represented by the equation y x 1. We want both of the equations, x y 5 and y = x 1, to be true for the same pair of numbers. The two equations are called a system of simultaneous equatio... |
in a coordinate plane using the same set of axes, the lines are parallel and fail to intersect, as in the case of x y 2 and x y 4. There is no common solution for the system of equations x y 2 and x y 4. It is obvious that there can be no ordered number pair (x, y) such that the sum of those numbers, x y, is both 2 an... |
that is, the graphs are the same line, every point on the line is a solution.The equations are consistent and dependent. 4. Check any solution by verifying that the ordered pair satisfies both equations. EXAMPLE 1 Solve graphically and check: 2x y 8 y x 2 Using a Graph to Solve a System of Linear Equations 413 Solutio... |
up arrow key to change to the second equation. Again note the coordinates of the point of intersection, (2, 4). ENTER: TRACE DISPLAY: Y1=–2X+8 * X=2 Y=4 Answer (2, 4), or x 2, y 4, or the solution set {(2, 4)} Using a Graph to Solve a System of Linear Equations 415 EXERCISES Writing About Mathematics 1. Are there any ... |
is 2. Find the numbers. 416 Writing and Solving Systems of Linear Functions 30. The sum of two numbers is 5. The larger number is 7 more than the smaller number. Find the numbers. 31. The perimeter of a rectangle is 12 meters. Its length is twice its width. Find the dimensions of the rectangle. 32. The perimeter of a ... |
a system of linear equations such as 2x y 2 and x y 2, we make use of the properties of equality to obtain an equation in one variable. When the coefficient of one of the variables in the first equation is the additive inverse of the coefficient of the same variable in the second, that variable can be eliminated by ad... |
4) Solve the resulting equation for the remaining variable, x: x 3y 13 [A] x y 5 1[B] 2y B] Check Substitute 1 for x and 4 for y in each of the given equations to verify the solution: x 3y 13 1 1 3(4) 5? 13 1 1 12 5? 13 13 13 ✔ x y 5 1 1 4 5? 5 5 5 ✔ Answer x 1, y 4 or (1, 4) Note: If equation [A] in Example 1 was x y ... |
[A] [B] Solution How to Proceed (1) Transform each of the given equations into equivalent equations in which the terms containing the variables appear on one side and the constant appears on the other side: (2) To eliminate y, find the least common multiple of the coefficients of y in equations [A] and [B]. That least... |
x 3y 27 15. 5r 2s 8 3r 7s 1 18. 4a 6b 15 6a – 4b 10 21. 3x 4y 2 x 2(7 y) 2a 1 1 1 3 2a 2 4 3b 5 8 3b 5 24 24. 4. a b 13 a b 5 7. a 4b 8 a 2b 0 10. 4x y 10 2x 3y 12 13. 2x y 26 3x 2y 42 16. 3x 7y 2 2x 3y 3 19. 2x y 17 5x 25 y 22. 3x 5(y 2) 1 8y 3x 25. c 2d 1 2 3c 5d 26 5. 3x y 16 2x y 11 8. 8a 5b 9 2a 5b 4 11. 5x 8y 1 3... |
in each account, solve the following system of equations: x y 600 0.03x 0.06y 9 30. Greta is twice as old as Robin. In 3 years, Robin will be 4 years younger than Greta is now. Let g represent Greta’s current age, and let r represent Robin’s current age. To find their current ages, solve the following system of equati... |
A] by replacing it with 2x 1: (2) Solve the resulting equation for x: (3) Replace x with its value in any equation involving both variables: (4) Solve the resulting equation for y: 4x 3y 27 y 2x 1 4x 3(2x 1) 27 4x 6x 3 27 10x 3 27 10x 30 x 3 y 2x 1 y 2(3) 1 y 6 1 y 5 [A] [B] [B→A] [B] Check Substitute 3 for x and 5 for... |
solving the equation 3x 4y 26 for y. Explain why the method used in Example 2 was easier. 2. Try to solve the system of equations x 2y 5 and y in the first equation. What conclusion can you draw? 1 2x 1 1 by substituting 1 2x 1 1 for y Using Substitution to Solve a System of Linear Equations 425 Developing Skills In 3... |
sides of the triangle. 28. A package of batteries costs $1.16 more than a roll of film. Martina paid $11.48 for 3 rolls of film and a package of batteries. a. Express the cost of a package of batteries, y, in terms of the cost of a roll of film, x. b. Write an equation that expresses the amount that Martina paid for t... |
find the numbers: x y 8.6 → 2(x y) 2(8.6) 2x 2y 17.2 3x 2y 6.3 23.5 5x x 4.7 The two numbers are 4.7 and 3.9. x y 8.6 4.7 y 8.6 y 3.9 Using Systems of Equations to Solve Verbal Problems 427 Procedure To solve a word problem by using a system of two equations involving two variables: 1. Use two different variables to r... |
13) $54 $78 $132. ✔ Answer A belt costs $6; a hat costs $13. EXAMPLE 2 When Angelo cashed a check for $170, the bank teller gave him 12 bills, some $20 bills and the rest $10 bills. How many bills of each denomination did Angelo receive? Solution (1) Represent the unknowns using two variables: Let x number of $10 bills... |
sum of two numbers is 74. The larger number is 3 more than the smaller number. Find the numbers. 5. The sum of two numbers is 104. The larger number is 1 less than twice the smaller number. Find the numbers. 6. The difference between two numbers is 25. The larger exceeds 3 times the smaller by 4. Find the numbers. 7. ... |
helpers earned $480. Each gardener receives the same pay for a day’s work and each helper receives the same pay for a day’s work. How much does a gardener and how much does a helper earn each day? 19. A baseball manager bought 4 bats and 9 balls for $76.50. On another day, she bought 3 bats and 1 dozen balls at the sa... |
5 pounds of eggplant is $5.41. What is the cost of one pound of squash, and what is the cost of one pound of eggplant? 27. One year, Roger Jackson and his wife Wilma together earned $67,000. If Roger earned $4,000 more than Wilma earned that year, how much did each earn? Graphing the Solution Set of a System of Inequa... |
set of the system x 2 and y 2 consists of the intersection of the solution sets of x 2 and y 2. Therefore, the dark colored region in the lower figure, which is the intersection of the graphs made in steps (1) and (2), is the graph of the solution set of the system x 2 and y 2 432 Writing and Solving Systems of Linear... |
) Using the same set of axes, graph y 2x 3 by first graphing the plane divider y 2x – 3. In the figure at the right, this is the solid line labeled m. The line y 2x 3 and the half-plane below this line together form the graph of the solution set of y 2x 3. (3) The dark colored region, the intersection of the graphs mad... |
are 7 cats, there can be 5 dogs. EXERCISES Writing About Mathematics 1. Write a system of inequalities whose solution set is the unshaded region of the graph drawn in Example 3. Explain your answer. 2. What points on the graph drawn in Example 3 are in the solution set of the system of open sentences x y 4 and y 2x 3?... |
�s class. 27. When Mr. Ehmke drives to work, he drives on city streets for part of the trip and on the expressway for the rest of the trip. The total trip is less than 8 miles. He drives at least 1 mile on city streets and at least 2 miles on the expressway. a. Write three inequalities to represent the information give... |
be shown on a graph as the intersection of the solution sets of the inequalities. VOCABULARY 10-4 System of simultaneous equations • Linear system • System of consistent equations • System of independent equations • System of inconsistent equations • System of dependent equations 10-5 Addition method 10-6 Substitution... |
by using addition to eliminate one of the variables. Check. 21. 2x y 10 x y 3 23. 3c d 0 c 4d 52 22. x 4y 1 5x 6y 8 In 24–26, solve each system of equations by using substitution to eliminate one of the variables. Check. 24 x 2y 7 x y 8 25. 3r 2s 20 r 2s 26. x y 7 2x 3y 21 In 27–32, solve each system of equations by u... |
Robby’s and Jason’s journeys to school. Who left first? Who arrived at school first? If each boy took the same route to school, when did Jason pass Robby? How fast did Robby walk? How fast did Jason ride his bicycle.25 1.0 0.75 0.50 0.25 0 ROBBY 8:00 8:05 8:10 8:15 8:20 8:25 8:30 TIME JASON CUMULATIVE REVIEW CHAPTERS ... |
) 2 (3) 3 (1) 6 (4) 56° (4) 3 10. When 2a2 5a is subtracted from 5a2 1, the difference is (1) 3a2 5a 1 (2) 3a2 5a 1 (3) 3a2 6a (4) 3a2 5a 1 Part II Answer all questions in this part. Each correct answer will receive 2 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, ... |
Draw a graph that shows the cost of becoming a member and using the photographic equipment. Let the vertical axis represent cost in dollars and the horizontal axis represent hours of use. b. On the same graph, show the cost of using the photographic equipment for non-members. c. When Tamara became a member of the Comm... |
their product. The numbers that are multiplied are factors of the product. Since 3(5) 15, the numbers 3 and 5 are factors of 15. Factoring a number is the process of finding those numbers whose product is the given number. Usually, when we factor, we are finding the factors of an integer and we find only those factors... |
, 21, 42, 63, and 126. Recall that a prime number is an integer greater than 1 that has no positive integral factors other than itself and 1. The first seven prime numbers are 2, 3, 5, 7, 11, 13, and 17. Integers greater than 1 that are not prime are called composite numbers. 444 Special Products and Factors In general... |
is 6a2b. Factors and Factoring 445 When we are expressing an algebraic factor, such as 6a2b, we will agree that: • Numerical coefficients need not be factored. (6 need not be written as 2 3.) • Powers of variables need not be represented as the product of several equal factors. (a2b need not be written as a a b.) EXAM... |
. 1 17. 128 22. 316 In 23–34, write all the positive integral factors of each given number. 23. 26 29. 37 24. 50 30. 62 25. 36 31. 253 26. 88 32. 102 27. 100 33. 70 28. 242 34. 169 35. The product of two monomials is 36x3y4. Find the second factor if the first factor is: a. 3x2y3 b. 6x3y2 c. 12xy4 d. 9x3y e. 18x3y2 36.... |
many common factors of 4rs and 8st such as 2, 4, 2s, and 4s. The greatest common monomial factor is 4s. We divide 4rs 8st by 4s to obtain the quotient r 2t, which is the second factor. Therefore, the polynomial 4rs 8st 4s(r 2t). 2. Factor 3x 4y. We notice that 1 is the only common factor of 3x and 4y, so the second fa... |
and b, 3a 5b is a prime number? Justify your answer. 3. 2a 2b Developing Skills In 3–29, write each expression in factored form. 4. 3x 3y 7. 4x 8y 10. 18c 27d 13. 6 18c 16. ax 5ab 19. 2x 4x3 22. pr2 2prh 25. 3ab2 6a2b 28. c3 c2 2c 6. xc – xd 9. 12x 18y 12. 7y 7 15. 2x2 5x 18. 10x 15x3 21. pr2 prl 24. 12y2 4y 27. 3x2 6... |
6b4) (6)(6)(b4)(b4) (6)2( b4)2 or 36b8 (4c2d3)2 (4c2d3)(4c2d3) (4)(4)(c2)(c2)(d3)(d3) (4)2(c2)2(d3)2 or 16c4d6 When a monomial is a perfect square, its numerical coefficient is a perfect square and the exponent of each variable is an even number. This statement holds true for each of the results shown above. Procedure ... |
4)2 2 A 14. 3 4 a B 24x2 18. 5 22. (0.06a2b)2 A B 2 Applying Skills In 23–28, each monomial represents the length of a side of a square. Write the monomial that represents the area of the square. 23. 4x 24. 10y 25. 2 3x 26. 1.5x 27. 3x2 28. 4x2y3 11-4 MULTIPLYING THE SUM AND THE DIFFERENCE OF TWO TERMS Recall that when... |
(4b)2 Write y2 49 9a2 16b2 EXERCISES Writing About Mathematics 1. Rose said that the product of two binomials is a binomial only when the two binomials are the sum and the difference of the same two terms. Miranda said that that cannot be true because (5a 10)(a 2) 5a2 20, a binomial. Show that Rose is correct by writi... |
multiplying the sum of two terms by the difference of the same two terms. Since the product (a b)(a b) is a2 b2, the factors of a2 b2 are (a b) and (a b). Therefore: a2 b2 (a b)(a b) Procedure To factor a binomial that is a difference of two perfect squares: 1. Express each of its terms as the square of a monomial. 2.... |
given polynomial represents the area of a rectangle. Express the area as the product of two binomials. 19. x2 4 23. 4x2 y2 22. t4 64 21. t2 49 20. y2 9 In 24–26, express the area of each shaded region as: a. the difference of the areas shown, and b. the product of two binomials. 24. c d c d d c 25. 2x 26. x d c 2x x y... |
: 1. Multiply the first terms of the binomials. 2. Multiply the first term of each binomial by the last term of the other bino- mial (the outer and inner terms), and add these products. 3. Multiply the last terms of the binomials. 4. Write the results of steps 1, 2, and 3 as a sum. Write the product (x 5)(x 7) as a tri... |
. (a 4)2 16. (7x 3)(2x 1) 19. (2x 5)2 22. (2t 3)(5t 1) 25. (2c 3d)(5c 2d) 28. (6t 1)(4t z) 3. (x 5)(x 3) 6. (8 c)(3 c) 9. (5 t)(9 t) 12. (y 8)2 15. (3x 2)2 18. (3x 4)2 21. (5y 4)(5y 4) 24. (5x 7y)(3x 4y) 27. (a b)(2a 3) Applying Skills 5. (x 10)(x 5) 8. (n 20)(n 3) 11. (c 5)(3c 1) 14. (2x 1)2 17. (2y 3)(3y 2) 20. (3t 2... |
the reverse of multiplying binomials of the form (dx e) and (fx g). When we factor a trinomial of this form, we use combinations of factors of the first and last terms. We list the possible pairs of factors, then test the pairs, one by one, until we find the pair that gives the correct middle term. For example, let us... |
binomial factors also has the opposites of these binomials as factors. Usually, however, we write only the pair of factors whose first terms have positive coefficients as the factors of a trinomial. 458 Special Products and Factors Procedure To factor a trinomial of the form ax2 bx c, find two binomials that have the ... |
) The product of the first terms of the binomials must be c2. Therefore, for each first term, we use c. We write: c2 5c 6 (c )(c ) (2) Since the product of the last terms of the binomials must be –6, one of these last terms must be positive and the other negative. The pairs of integers whose product is –6 are: (1)(6) (... |
last terms must be positive and the other negative. The pairs of integers whose product is –15 are: (1)(15) (1)(15) (3)(5) (3)(5) (3) These four pairs of integers will form eight pairs of binomial factors since there are two ways in which the first terms can be arranged. Note how (2x 1)(x 15) is not the same product a... |
c2 2c 35 27. 3x2 10x 8 31. 10a2 9a 2 4. c2 6c 5 8. y2 6y 8 12. y2 2y 8 16. a2 11a 18 20. x2 x 2 24. x2 7x 18 28. 16x2 8x 1 32. 3a2 7ab 2b2 5. x2 8x 7 9. y2 9y 8 13. y2 7y – 8 17. z2 10z 25 21. x2 6x 7 25. z2 9z 36 29. 2x2 x 3 33. 4x2 5xy 6y2 6. x2 11x 10 10. y2 9y 8 14. y2 7y 8 18. x2 5x 6 22. y2 4y 5 26. 2x2 5x 2 30.... |
factor is 1).Then: • Factor any trinomial into the product of two binomials if possible. • Factor any binomials that are the difference of two perfect squares as such. 3. Write the given polynomial as the product of all of the factors. Make certain that all factors except the monomial factor are prime polynomials. EXA... |
pd2 19. x3 7x2 10x 22. 2ax2 2ax 12a 25. y4 13y2 36 28. 16x2 16x 4 3. 2a2 2b2 6. st2 9s 9. 18m2 8 12. z3 z 15. y4 81 18. 4r2 4r 48 21. d3 8d2 16d 24. a4 10a2 9 27. 2a2b 7ab 3b 5. ax2 ay2 8. 3x2 27y2 11. x3 4x 14. x4 1 17. 3x2 6x 3 20. 4x2 6x 4 23. 16x2 x2y4 26. 5x4 10x2 5 29. 25x2 100xy 100y2 464 Special Products and F... |
Write the given polynomial as the product of these factors. VOCABULARY 11-1 Product • Factoring a number • Factoring over the set of integers • Greatest common factor 11-2 Factoring a polynomial • Common monomial factor • Greatest common monomial factor • Prime polynomial 11-3 Square of a monomial 11-5 Difference of t... |
(1) a2y 10ay 25y (2) 2ax2 2ax 12a (3) 18m2 24m 8 (4) c2z2 18cz2 81z2 34. Of the four polynomials given below, explain how each is different from the others. x2 9 x3 5x2 6x 35. If the length and width of a rectangle are represented by 2x 3 and 3x 2, respectively, express the area of the rectangle as a trinomial. x2 2x ... |
, at the same rate, of 7 pounds of apples? (1) $7.25 (2) $8.25 (3) $8.50 (4) $8.75 Cumulative Review 467 (1) a2 4a 4 2. When 3a2 4 is subtracted from 2a2 4a, the difference is (2) a2 4a 4 (3) a2 8a 3. The solution of the equation x 3(2x 4) 7x is (4) a2 a (1) 6 (2) 6 (3) 1 (4) 1 4. The area of a circle is 16p square cen... |
68. Find, to the nearest degree, the measure of the smallest angle of the triangle. 12. Solve and check: x x 1 8 5 2 3. 468 Special Products and Factors Part III Answer all questions in this part. Each correct answer will receive 3 credits. Clearly indicate the necessary steps, including appropriate formula substituti... |
.A tiny error can send the craft thousands of miles off course.Why do such errors occur? Space centers rely heavily on sophisticated computers, but computers and calculators alike work with approximations of numbers, not necessarily with exact values. We have learned that irrational numbers, such as 2 and 5 ", are show... |
are rational numbers; others, such as " and radicals that are rational numbers. 4 " and Radicals and the Rational Numbers 471 Perfect Squares Any number that is the square of a rational number is called a perfect square. For example, 3 3 9 0 0 0 1.4 1.4 1.96 2 7 3 2 7 5 4 49 Therefore, perfect squares include 9, 0, 1.... |
number, the negative sign must be entered before the radical. ENTER: (-) 2nd ¯ 25 ENTER DISPLAY: - √ ( 2 5 - 5 The calculator will display an error message if it is set in “real” mode and the square root of a negative number is entered. ENTER: 2nd ¯ (-) 25 ENTER DISPLAY Cube Roots and Other Roots A cube root of a numb... |
In 3–22, express each radical as a rational number with the appropriate signs. 16 3. 8. " 2 " 625 4. 9. 64 2 " 1 4 $ 6 5. 6 2 10. " 100 9 16 $ 13. 2 1.44 14. 0.09 15. 6 0.0004 18. " 5 32 " " 3 28 " In 23–32, evaluate each radical by using a calculator. " 3 2125 " 19. 20. 2 6. 6 169 11. 16. 21. " 6 25 81 $ 3 1 " 4 0.12... |
in each case: a. the length of each side of a square that has the given area b. the perimeter of the square. 60. 36 square feet 61. 196 square yards 62. 121 square centimeters 64. Express in terms of x, (x 0), the perimeter of a square whose area is represented by x2. 63. 225 square meters 65. Write each of the intege... |
nearest millionth) rounded to five decimal places (nearest hundred-thousandth) rounded to four decimal places (nearest ten-thousandth) rounded to three decimal places (nearest thousandth) rounded to two decimal places (nearest hundredth) is greater Each rational approximation of than 2 but less than 3. This fact can be... |
or more steps, no radical should be rounded until the very last step. For example, to find the value of 3 3 first multiply the calculator approximation for product to two decimal places. 5 " 5, rounded to the nearest hundredth, " by 3 and then round the Correct Solution: Incorrect Solution(2.236067977) 6.708203931 6.7... |
Since (2.84)2 8.0656, then 8.0656 2.84. " Answer 8.0656 " is a rational number. If n is a positive rational number written as a terminating decimal, then n2 has twice as many decimal places as n. For example, the square of 2.84 has four decimal places. Also, since the last digit of 2.84 is 4, note that the last digit ... |
. " " In 13–18, in each case, write the given numbers in order, starting with the smallest. " " " 13. 2, 16. 0 " 14. 4, 17. 5, 17, 3 " 21, " 30 " 15. 18. In 19–33, state whether each number is rational or irrational. 19. 24. 29. 25 " 400 1,156 " " 20. 25. 30. 40 " 1 2 $ 951 " 21. 26. 31. 2 2 36 " 4 9 $ 6.1504 " 22. 27.... |
19 12-3 FINDING THE PRINCIPAL SQUARE ROOT OF A MONOMIAL Just as we can find the principal square root of a number that is a perfect square, we can find the principal square roots of variables and monomials that represent perfect squares. • Since (6)(6) 36, • Since (x)(x) x2, • Since (a2)(a2) a4, • Since (6a2)(6a2) 36a4... |
represent positive numbers. 3. 4a2 7. 9c2 4. 49z2 8. 36y4 5. 16 25r2 9. c2d2 6. 0.81w2 10. 4x2y2 11. 144a4b2 12. 0.36m2 13. 0.49a2b2 14. 70.56b2x10 Applying Skills In 15–18, where all variables represent positive numbers: a. Represent each side of the square whose area is given. b. Represent the perimeter of that squa... |
and is a fraction, change it to an equivalent fraction that has a denominator that is a perfect square. Write the radicand as the product of a fraction that is a perfect square and an integer that has no perfect-square factor other than 1. For example: 8 3 5 $ Procedure 3 3 3 8 3 5 $ 24 " To simplify the square root of... |
10. 15. 20. 63 " 1 4 " 48 32 3 $ 49x5 " 6. 11. 16. 21. 98 96 " 3 4 " 2 1 8 $ 36r2s " 7. 4 12 " 80 3 2 " 15 12. 17. 22. 2 5 $ 243xy2 " (3) 4 3 " (3) 32 " (3) 9 2 " (3) 12 " (4) 16 3 " (4) 64 " (4) 3 6 " (4) 27 " In 27–30, for each expression: a. Use a calculator to find a rational approximation of the expression. b. Wr... |
(10) 5 10 (6 2 1) 5 " 3 7(5) 5 5 " 3 7 " Procedure To add or subtract like terms that contain like radicals: 1. Add or subtract the coefficients of the radicals. 2. Multiply the sum or difference obtained by the common radical. 488 Operations With Radicals Adding and Subtracting Unlike Radicals Unlike radicals are radi... |
25 25 50 CE2 CD2 DE2 32 32 9 9 18 CE 18 " 2 5 5 18 5 25? " " " AC 50 " 50 1 " " b. AC CE c. Error 8 2 2 11.25 " Percent of error 8 " 2 2 11.25 2 8 " ¯ 2 8 2nd 8 2nd ¯ 2 ENTER: ( ( ) ) ) 11.25 ) ENTER DISPLAY Multiply the number in the display by 100 to change to percent. Percent of error 0.5631089% 0.56% Answers a. AC... |
and 29: a. Express the perimeter of the figure in simplest radical form. b. Using a calculator, approximate the expression obtained in part a to the nearest thousandth. Multiplication of Square-Root Radicals 491 28. 29. 5√5 4√5 3√5 √27 2√3 30. On the way to softball practice, Maggie walks diagonally through a square f... |
(x 0). 9x2? " " 2 5 3x Answer 2 " Multiplication of Square-Root Radicals 493 EXERCISES Writing About Mathematics 1. When a and b are unequal prime numbers, is answer. rational or irrational? Explain your ab " 2. Is 4a2 a rational number for all values of a? Explain your answer. " Developing Skills In 3–26: in each cas... |
ROOT RADICALS Since and then " " $, Since and then. 4 9, 16 25 5 4 5 16 5 4 5 25 16 25 5 " " $ " " $ 16 25. These examples illustrate the following property of square-root radicals: The square root of a fraction that is the quotient of two positive numbers is equal to the square root of the numerator divided by the squ... |
8. 12. 16. 75 4 " 24 4 " 7 " 20 4 50 " 2 " 5. 9. 13. 17. 70 4 " 10 " 150 " b3c4 a2 a " " 6. 10. 14. 18. 14 4 2 " 21 " 40 4 " 9y 4 " 3 x3y " z 6 " 5 " y " In 19–26, state whether each quotient is a rational number or an irrational number. 19. 5 " 7 23. 9 16 " " 20. 24. 50 2 18 25 " " " " 21. 25. 18 3 " " 25 " 5 " 24 2 ... |
number. Principal Square Root Negative Square Root Both Square Roots x2 5 ZxZ " 2 x2 5 2ZxZ " 6 x2 5 6x " Finding the cube root of a number is the inverse of the operation of cubing. if and only if because 43 = 4 4 4 64. In general, 3 b 5 x " 3 64 5 4 Thus, " x3 b. and 7 " Like radicals have the same radicand and the ... |
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