Chapter 1 Questions - La Sierra...

Chapter 1 Questions

1. Multiply the complex numbers and write the answer in standard form.

(1− 5i)(5 + 3i)

1. (a) Simplify i403 and write answer as a complex number in standard form.

(b) Simplify5

i283and write answer as a complex number in standard form.

1. Simplify the following fraction and write it as a complex number in standard form.

2 + 3i

2− 4i

1. Solve the equation7(3s− 5)− 4(8s− 8) = −25

1. Solve the following equation12

4x− 5=

4x + 7

4x− 5+ 2

1. Solve the absolute value equation, or indicate that there is no solution.

6− 5|1− 3x| = −24

1. Suppose the final test in a class is worth 30 percent of the overall grade, and assignmentsare worth 15 percent of the overall grade and term tests are worth the remaining 55 percent ofthe grade. If a student has an average of 97% on homework and 87% on term tests going intothe final. What percentage is needed on the final test for the student to finish with an overallaverage of 90%. Express answer to nearest whole number.

1. Ruben is driving along a highway that passes through Barstow. His distance d, in miles,from Barstow is given by the equation d = |130− 55t|, where t is the time, in hours, since thestart of his trip and 0 ≤ t ≤ 16. When will Ruben be exactly 75 miles from Barstow? Expressyour answers in hours, accurate to two decimal places.

1. Identify each of the equations below as a conditional equation, identity or contradiction.

(a) 6(5x + 3)− 30x = 18

(b) 6(5x + 3) = 19 + 30x

(c) 3(5− 3x)− 5x = 16− 15x

1. Many physical or mathematical formulas involve combinations of products and sums of thevariables. Here is a random example to give you practice with that type of situation whereyou are to solve the following equation for C

A =1

5H(B + 3C)

1. Many physical formulas involve reciprocals of the variables. Here is a random exampleto give you practice with that type of situation where you are asked to solve the followingequation for x

8

x+

9

y=

8

z

1. A retailer determines the price of a dress by first computing 155% of the wholesale price ofthe dress and then adding a markup of $8.00. What is the wholesale price of a dress that hasa retail price of $141.30?

1. A worker can paint a house in 23 hours. With the help of an assistant, the house can bepainted in 17 hours. How long would it take the assistant to paint the house alone? Expressfinal answer to nearest 0.1 hour.

1. An investment adviser invested $13500 in two accounts. One investment earned 3% annualsimple interest, and the other investment earned 5% annual simple interest. The amount ofinterest earned for one year was $541. How much was invested in each account?

1. A tea merchant wants to make 220 pounds of blended tea costing $7.00-per-pound. Theblend is made using a $5.85-per-pound grade of tea and a $8.60-per-pound grade of tea. Howmany pounds of each grade of tea should be used?

1. Jogging at an average rate of 5 meters per second, a runner ran to the end of a path. Therunner then jogged back to the starting point at an average rate of 4 meters per second. Thetotal time for the jog to the end of the path and back was 2 minutes and 42 seconds (or a totalof 162 seconds). Find the distance from the point the runner started to the end of the path.

1. The length of a rectangle is 9 feet less than 4 times the width of the rectangle. If theperimeter of the rectangle is 92 feet, find the width and the length of the rectangle.

1. A goldsmith has two gold alloys. The first alloy is 30% gold; the second alloy is 46% gold.Set-up and solve a linear equation to determine how many grams of each alloy should be mixedto produce 40 grams of an alloy that is 36% gold?

1. Solve the quadratic equation 3x2 + 11x = 4.

1. Solve the quadratic equation 3x2 + 3x = 4 using the quadratic formula.

1. Solve the quadratic equation x2 = 4x− 17 using the quadratic formula.

1. Find the discriminant, and use it to determine whether1

5x2 − 7

5x = 2 has (i) two distinct

real solutions; (ii) one real solution; or (iii) two complex (nonreal) solutions. You do not neetto find the solutions.

1. A television screen measures 162 cm diagonally and its aspect ratio is 16 to 9. Find thedimensions of the television screen. Round your answers to the nearest tenth of a cm.

1. A square piece of cardboard is formed into a box by cutting out 3-inch squares fromeach of the corners and folding up the sides, as shown in the following figure (not to scale). If

the volume of the box needs to be 1323 cubic inches, what size square piece of cardboard isneeded?

1. A gardener wishes to use 520 feet of fencing to enclose a rectangular region and subdividethe region into two smaller rectangles. See the figure below. The total enclosed area is 11200square feet. Find the possible width(s) of the enclosed area. Round answer(s) to the nearest0.1 feet.

1. A model rocket is launched upward with an initial velocity of 230 feet per second. Theheight, in feet, of the rocket t seconds after the launch is given by h(t) = −16t2 + 230t. Howmany seconds after the launch will the rocket be 270 feet above the ground? Round answer tonearest hundredth of a second.

1. Find all real solutions to the equation 3x4 + 4x2 = 15.

1. Find all solutions to the equation 3x2/3 − 26x1/3 − 77 = 0.

1. Solve the equation 2x3 − 3x2 − 6x + 9 = 0 by factoring.

1. Solve the rational equation x− 4x + 6

x− 3=

3x− 27

x− 3.

1. Use algebra to solve the equation x−√x + 10 = 10.

1. Use algebra to solve√

2x + 51−√x + 10 = 4. Check all proposed solutions.

1. Working together in parallel, two computers can solve a problem in 150 minutes. Workingalone, the faster computer can solve the problem in 55 minutes less than the slower computer.How long would it take the faster computer to solve the problem working alone?

1. A car and a truck are making a long trip on the same route. The car travels at a constantrate 18 km/h faster than the truck that also travels at a constant rate. The truck started thetrip half-an-hour before the car. The car overtook the truck after traveling 180 km. What isthe rate of each vehicle?

1. Solve the inequality |4x− 4| > 20. Use interval notation to express the solution set.

1. Solve the inequality | − 3x + 6| ≤ 6. Use interval notation to express the solution set.

1. Solve the quadratic inequality x2 − 6x ≥ 55.

1. Solve the inequalityx + 5

x− 9≥ 2 and write the answer in interval notation.

1. Solve the inequality11− x

x− 5≥ −2 and write the answer in interval notation.

1. Solve the inequalityx + 5

x− 4> 4 and write the answer in interval notation.

1. Solve the inequality(x + 4)(x− 9)

x + 5≤ 0 using the method of critical values with test

numbers or a sign chart. Write your answer in interval notation.

1. Suppose the final test in a class is worth 30 percent of the overall grade, and assignmentsare worth 15 percent of the overall grade and term tests are worth the remaining 55 percentof the grade. If a student has an average of 77% on homework and 75% on term tests goinginto the final. Determine the interval of percentages needed on on the final test for the studentto finish with an overall percentage between 65% and 75%. Express endpoints of interval to 1decimal place..

1. (A physics inequality) The equation s = −16t2 + v0t + s0 gives the height s, in feet aboveground level, at the time t seconds, of a projectile launched directly upward from a height s0feet above the ground and with an initial velocity of v0 feet per second. A catapult launchesa rock directly upward from an initial a height of 15 feet above the ground with an initialvelocity of 112 feet per second. Find the time interval during which the rock will be more than111 feet above the ground.

Hint. This will reduce to a quadratic inequality where all terms have a factor of 16.

1. The area, A, of a picture projected on a movie screen varies directly as the square of thedistance, d, from the projector to the screen.

(a) Write an equation that expresses the relationship between the variables. Use k as theconstant of variation.

(b) If a distance of 16 feet produces a picture with an area of 43 square feet, what distanceproduces a picture with an area of 1075 square feet.

1. The speed of a bicycle gear, in revolutions per minute, is inversely proportional to thenumber of teeth on the gear. If a gear with 80 teeth has a speed of 30 revolutions per minute,what will be the speed of a gear with 60 teeth?

1. The volume V of a right circular cone varies jointly as the square of the radius r and theheight h.

(a) Write an equation representing the relation between the given variables. Use k as thevariation constant.

(b) Determine what happens to V when the radius is tripled.

(c) Determine what happens to V when the radius is tripled and the height is quadrupled.

1. (a) The maximum load L that a cylindrical column of circular cross section can supportvaries directly as the fourth power of diameter d and inversely as the square of its heighth. Write an equation that represents the relation between the given variables. Use k as thevariation constant.

(b) If a column 3 feet in diameter and 18 feet high supports up to 18 tons, use the relationin (a) to determine how much a column 2 feet in diameter and 14 feet high made of the samematerial can support. Round your answer to the nearest hundredth of a ton.

1. (a) The load L a horizontal beam can safely support varies jointly as the width w and thesquare of the depth d and inversely as the length l. Write an equation that represents therelation between the given variables. Use k as the constant of variation.

(b) If a 16-foot beam with width 10 inches and depth 11 inches safely supports 550 pounds, howmany pounds can a 13-foot beam that has width 12 inches and depth 13 inches be expected tosupport? Round answer to the nearest pound. Assume the two beams are made of the samematerial.

1. The force F needed to keep a car from skidding on a curve varies jointly as the weight w ofthe car and the square of its speed v and inversely as the radius of the curve r.

(a) Write an equation that represents the relationship between the variables (use k as theconstant of variation).

(b) Suppose it takes 3300 pounds of force to keep a 2800 pound car from skidding on a curvewith radius 420 feet at 50 miles per hour. What force is needed to keep the car from skiddingwhen it takes a similar curve with radius 435 feet at 65 miles per hour? Round to the nearest10 pounds.

Chapter 2 Problems

2. Find the midpoint of the line segment with the given endpoints.

(−4, 10), (−6, 2).

2. Find the other endpoint of the line segment that has the given endpoint and midpoint.

Endpoint (−9, 1), Midpoint (−5,−2).

2. Find the distance between the given points. (You are not required to simplify the answer)

(5, 3) and (−5, 7)

2. A circle has a diameter with endpoints (−5, 10) and (−7, 2). Find the equation of the circlein standard form. (Hint: the center is the midpoint of the endpoints of the diameter).

2. Write the equation of the circle x2 + y2 + 6x− 6y + 17 = 0 in standard form. Then findthe center and radius of the circle.

2. Find the x- and y-intercepts of the graph of the following equation.

15x + 1 = y2

2. Use the graph of the function f given below to answer the following questions.

(a) Find the x-intercept(s), if any, of the graph of f .

(b) Find the y-intercept(s), if any, of the graph of f .

(c) Find f(−10)

(d) Find f(−5)

(e) Find f(5)

(f) Find f(10)

(g) Find f(15)

−20−15−10 −5 5 10 15 20

−20

−15

−10

−5

5

10

15

20

x

y

2. Let f(z) = 2z2 − 2z. Find the following

(a) f(−4) (b) f(3) (c) f(3 + h) (d) f(x + h)

2. Consider the piecewise defined function f(x) =

5− 2x, if x ≤ −6;

2 if − 6 < x < 5;

x2 − 5 if x ≥ 5.

Find:

(a) f(−11) (b) f(−6) (c) f(−5) (d) f(5) (e) f(t + 2) for t ≥ 3

2. Consider the function f defined by

f(x) =x− 2

3x + 4

(a) Find the domain of f . Write your answer in interval notation.

(b) Is 3 in the range of f? If so, find all x so that f(x) = 3.

(c) Is1

3in the range of f? If so, find all x so that f(x) =

1

3.

2. Find the zeros of the function f given below; that is find all x so that f(x) = 0.

f(x) = 8x3 + 5x2 − 40x− 25.

2. Let y = 1− |x + 2|.

(a) Complete the following table.

x −6 −5 −4 -3 -2 -1 0 1 2

y

(b) Plot the graph of y = 1− |x + 2| and determine its x-intercepts and y-intercepts.

2. (a) Find the domain of the function

f(x) =√x + 4− 5

(b) Sketch the graph of f(x) =√x + 4− 5 by plotting appropriate points.

2. Determine intervals on which the function graphed below is: (a) increasing; (b) decreasing;(c) constant.

−6 −4 −2 2 4 6

−6

−4

−2

2

4

6

x

y

2. Two graphs are given below. For each graph, determine whether it is a graph of a function,and if it is, determine whether the function is be one-to-one. Explain your answers.

(a) (b)

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

x

y

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

x

y

2. An open box is to be made from a square piece of cardboard having dimensions 98 cm by98 cm (w = 98) by cutting out squares of area x2 from each corner, as shown in the figurebelow.

(a) Express the volume V (in cubic centimeters) of the boxas a function of x.

(b) State the domain of V

2. An athlete swims from point A to point B at a rate of 75

mph and runs from point Bto point C at a rate of 11 mph. Use the dimensions in the figure below to write the time trequired to reach point C as a function of x.

2. Answer the following questions concerning the line that is graphed below.

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

x

y

(a) Find the coordinates of the y-intercept.

(b) Find the coordinates of the x-intercept.

(c) Find the slope of the line.

(d) Write the equation of the line in slope-intercept form.

(e) Find the equation of the line parallel to the given line that passes through the origin, andgraph the line on the same graph.

2. Find the equation of the line through the points (4, 4) and (3, 7). Write the equation inslope-intercept form.

2. (a) A line passing through the point (−1, 2) has slope m = −5. Find the equation of theline in slope intercept form.

(b) Find the slope and y-intercept of the line passing through the point (10, 1) that is perpen-dicular to the line described in part (a).

(c) Graph both lines on the same graph.

2. A magazine company had a profit of $44400 per year when it had 14000 subscribers. Whenit obtained 16000 subscribers, it had a profit of $56400. Assume the profit P is a linear functionof the number of subscribers s.

(a) Find the function P .

(b) What will the profit be if the company obtains 28000 subscribers?

(c) What is the number of subscribers to break even? (Round to the next highest subscribernumber if the number is not whole).

2. Julie opened a lemonade stand and found that daily her profit is a linear function of thenumber of cups of lemonade sold. When she sells 300 cups of lemonade, she makes $15 andwhen she sells 400 cups of lemonade, she makes $30.

(a) Find the profit function.

(b) How many cups of lemonade does Julie need to sell to break even on a given day?

(c) How many cups of lemonade does Julie need to sell to make $60 in a day?

(d) How much would Julie make on a day when she sells 1000 cups of lemonade?

2. A quadratic function f is graphed below. It is a translation of the graph of y = −3x2. Usethe graph to answer the following questions.

(a) Find the vertex of the quadratic function f .

(b) What is the axis of symmetry of the graph of f?

(c) Does f(x) have a maximum value? If so, at what value of x does it occur, and what is themaximum value?

(d) Does f(x) have a minimum value? If so, at what value of x does it occur, and what is themaximum value?

(e) Find the range of f(x). Express your answer in interval notation.

(f) Find f(x), that is, write the expression quadratic function f .

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

x

y

2. For this problem, f is the quadratic function

f(x) = 2x + 4x + 8

(a) Find the vertex of the graph of f .

(b) Write the quadratic function f(x) in standard form.

(c) Does the graph of f open upward or downward?

(d) Does f have a maximum value? If so, what is it?

(e) Does f have a minimum value? If so, what is it?

(f) Find the range of f . Write your answer in interval notation.

2. (a) Find the coordinates of the vertex of the graph of the quadratic function f defined by

f(x) = −2x2 + 9x + 1

(b) Find the range of the quadratic function f from (a). Express your answer in intervalnotation.

(c) Write the quadratic function f in standard form.

2. The height above the ground, in feet, of a projectile launched with an initial velocity of 32feet per second from an initial height of 15 feet above the ground is a function of time t inseconds, given by

h(t) = −16t2 + 32t + 15.

(a) Find the time t when the projectile reaches its maximum height.

(b) Find the maximum height attained by the projectile.

(c) Find the time t when the projectile hits the ground (has a height of 0 feet). Express answerin seconds to the nearest decimal place.

2. A farmer has 888 feet of fencing with which to make a rectangular enclosure that will besubdivided into two separate enclosures (see the figure below).

(a) Write the length l as a function of width w.

(b) Write the total area as a quadratic function of w.

(c) Find the dimensions of the enclosure that will produce the greatest enclosed area.

2. Determine whether the following functions are even, odd, or neither.

(a) f(x) = 8x9 + 2x− 5

(b) g(x) = 9x8 − 2x2 − |x|+ 7

(c) h(x) = 2x9 − 7x3 + x

2. The graph of two different functions are given below in (a) and (b).

(a) (b)

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

x

y

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

x

y

(i) Is the function graphed above in (a) even, odd, or neither? Explain your answer.

(ii) Is the function graphed above in (b) even, odd, or neither? Explain your answer.

2. Which of the symmetries (over x-axis, y-axis, origin) are possessed by the graphs of thefollowing equations? Explain your answers.

(a) x2 + |y| = y11.

(b) 8|x|3 − 2|y|3 = −7.

(c) y9 = 2x + 2.

(d) 2y + 11x = 0.

2. The graph of f(x) = x3 is given below. Find the function g(x) whose graph is the graph off shifted horizontally 4 units to the right and vertically 2 units up, and then graph g on thegraph below.

−6 −4 −2 2 4 6

−6

−4

−2

2

4

6

f(x)

x

y

2. The graph of the function f given below was obtained from horizontal and vertical trans-lations of the graph of y =

√x. Find the function f(x).

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

x

y

For your reference, the points (−3,−3), (−2,−2), (1,−1), and (6, 0) were plotted on the graphof f .

2. The graph of y = f(x) is given below in (a) and (b).

(a) (b)

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

x

y

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

x

y

On the graph of (a) above, sketch y = f(x + 3), and on the graph (b) above, sketch y =f(x + 3)− 4.


(a) (b)

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

x

y

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

x

y

On the graph of (a) above, sketch y = −f(x), and on the graph (b) above, sketch y = f(−x).


(a) (b)

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

x

y

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

x

y

(i) On the graph of (a) above, sketch the graph of y = 2f(x).

(ii) On the graph of (b) above, sketch the graph of y = 2f(x)− 1.

2. The graph of y = |x| along with another graph is given in each of the graphs below in (a)and (b). In each case, the other graph is a result of reflections and translations of y = |x|.

(a) (b)

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

x

y

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

x

y

(i) Find the equation of the other graph in (a).

(ii) Find the equation of the other graph in (b).

2. The graph of g in (a) was obtained from reflections of the graph of y =√x, the graph of

h in (b) was obtained by horizontal and vertical translations of the graph of g from (a). Findthe functions h and g. (The points (0, 0) and (2, 3) are indicated on the graphs of g and hrespectively for your reference.)

(a) (b)

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

g

x

y

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

h x

y

2. Let f(x) = 3x7 − 4 and g(x) =

(x + 4

3

) 17

.

(a) Find (f ◦ g)(x).

(b) Find (g ◦ f)(x).

2. Let f(x) =√

2x + 1 and g(x) =√

4− x. Find the domains of f + g, f − g, fg andf

g.

Write your answers in interval notation.

2. Let f(x) = 4x2 − 5x + 4 and g(x) = 4x− 4.

(a) Find (f ◦ g)(x)

(b) Find (g ◦ f)(x)

2. Let f(x) = 5x2 − 6 and g(x) = |x− 2|, find

(a) (g ◦ f)(−4) (b) (f ◦ g)(−4) (c) (f ◦ g)(0) (d) (fg)(1) (e) (f + g)(1)

2. Write the definition of the piecewise defined function f whose graph is given below.

−6 −4 −2 2 4 6

−6

−4

−2

2

4

6

x

y

2. Let f(x) = 2x2 − 2x + 4. Find and simplify the difference quotientf(x + h)− f(x)

h.

Chapter 3 Problems

3. Use long division to find

(6x3 − 6x2 − 4x + 4)÷ (2x2 + 5).

3. Use synthetic division to find (x4 + 3x3 − 7x + 10)÷ (x + 4)

3. Use synthetic division to find (3x5 − 7x3 + 2x2 − 6x + 2)÷ (x− 3)

3. Consider the polynomials P and Q defined as follows

P (x) = x16 + 5x− 65539 and Q(x) = x16 + 8x− 65520

(a) Use a calculator to find P (−2), P (2), Q(−2) and Q(2).

(b) Use your answers from (a) and the remainder and factor theorems to answer the followingquestions.

• What is the remainder of P (x)÷ (x− 2)? Is (x− 2) a factor of P (x)?

• What is the remainder of P (x)÷ (x + 2)? Is (x + 2) a factor of P (x)?

• What is the remainder of Q(x)÷ (x− 2)? Is (x− 2) a factor of Q(x)?

• What is the remainder of Q(x)÷ (x + 2)? Is (x + 2) a factor of Q(x)?

3. Ken was trying to factor a polynomial, so he programmed the formula for P (x) in hiscalculator and he correctly found that

P (−8) = −22, P (−5) = 0, P (−1) = 0,P (0) = −13, P (3) = 0, and P (8) = 17.

(a) Even with this information Ken was still puzzled, so he asked his brilliant girlfriend JoAnn,and she gave him a hint by telling him that there are three obvious factors from the giveninformation. Is JoAnn correct? If so, what are the factors?

(b) Based on the values of P given above. Answer the following questions.

(i) What is the remainder of P (x)÷ (x + 8)?

(ii) What is the remainder of P (x)÷ (x + 5)?

(iii) What is the remainder of P (x)÷ x?

(iv) What is the remainder of P (x)÷ (x− 8)?

3. Use synthetic division to show that (x + 8) is a factor of the polynomial P given by

P (x) = x3 + 11x2 + 25x + 8

Then write P as the product of (x + 8) and a quadratic factor.

3. Determine the far right and far left behavior of the four polynomials given below

P (x) = −4x11 + 400x10 − 2x3 + 1x− 3.

Q(x) = 4x11 + 400x10 − 2x3 + 1x− 3.

R(x) = −4x12 + 400x10 − 2x3 + 1x− 3.

S(x) = 4x12 + 400x10 − 2x3 + 1x− 3.

For each of the polynomials above, state whether its leading coefficient is negative or positive,and state whether its degree is even or odd, and then choose one of the following options.

(a) Up to far left and up to far right

(b) Up to far left and down to far right

(c) Down to far left and down to far right

(d) Down to far left and up to far right

(e) None of the above.

3. Use the Intermediate Value Theorem to determine whether P has a zero between a and b.

P (x) = 3x3 + 7x2 − 5x− 3; a = 1, b = 3

First, find P (1) = and P (3) = . Then choose the best responsefrom the following:

(a) Because P (1) and P (3) have opposite signs, we know that P has at least one real zerobetween 1 and 3.

(b) Because P (1) and P (3) have opposite signs, we do not know if P has at least one realzero between 1 and 3.

(c) Because P (1) and P (3) have the same sign, we know that P has at least one real zerobetween 1 and 3.

(d) Because P (1) and P (3) have the same sign, we do not know if P has at least one realzero between 1 and 3.

3. Find the real zeros of the polynomial function by factoring; for this polynomial try factoringby grouping.

P (x) = x3 − 2x2 − x + 2

3. Determine the x-intercepts for the graph of P . For each x-intercept, use the Even and OddPowers of (x−c) Theorem to determine whether the graph of P crosses the x-axis or intersectsbut does not cross the x-axis at that intercept.

P (x) = (x2 − 16)2(x + 6)5(x− 8)7

3. Use the polynomial P given below in both standard and factored form to answer thefollowing questions.

P (x) = 2x3 + 2x2 − 10x + 6 = 2(x− 1)2(x + 3)

(a) Determine the far right and far left behavior of P .

(b) List the x-intercepts, and at each intercept determine whether the graph of P crosses orintersects but does not cross the x-axis.

(c) Find the y-intercepts.

(d) Use the above information to sketch a rough graph of P .

3. Find the zeros of the polynomial function, and state the multiplicity of each zero.

P (x) = x5(16x− 5)6(x2 − 25)2

3. Use the graph of the polynomial P given below to answer the following questions.

−3 −2 −1 1 2 3 4 5

−5−4−3−2−1

12345

x

y

(a) Use the far right and far left behavior to determine whether the degree of P is even or odd.

(b) Use the far right and far left behavior to determine whether the leading coefficient of P ispositive or negative.

(c) List the zeros of P and for each zero, determine whether it has even multiplicity or oddmultiplicity.

3. Use the rational zero theorem to determine the possible rational zeros of

P (x) = 14x10 − 5x6 + 7x2 + 7x− 6

3. Use Descartes’ Rule of Signs to state the number of possible positive and negative real zerosof the given polynomial functions.

(a) P (x) = 7x5 + 2x4 + 2x3 + x2 + 4x− 2

(b) Q(x) = 4x6 − 7x5 − x4 + 4x3 − 2x2 + 2x + 70

3. Find the zeros of the polynomial P by using synthetic division to divide out one rationalzero of P , and then solve the remaining quadratic to find the remaining two zeros

P (x) = x3 + 3x2 − 17x− 3

3. Find the zeros of the polynomial P by using synthetic division to successively divide outtwo rational zeros of P and then solve the remaining quadratic to find the remaining two zeros.

P (x) = x4 + 12x3 + 48x2 + 67x + 12

3. Find the zeros of the polynomial P by using synthetic division to successively divide outtwo rational zeros of P and then solve the remaining quadratic to find the remaining two zeros

P (x) = x4 + 6x3 + 4x2 − 9x− 2

3. Given that 2 + i is a zero of Q(x) = x4 − 4x3 + 9x2 − 16x + 20 find the remaining zeros,and write Q(x) as a product of linear factors.

3. Find all zeros of the polynomial P defined by

P (x) = x4 − 3x3 + 2x2 − 3x + 1

given that x = i is a zero of P .

3. (a) Find a polynomial P with real coefficients of smallest degree that has zeros 2i, and −1.

(b) Find a polynomial P with real coefficients of smallest degree that has zeros 2i, and −1such that P (0) = 8.

3. Write P (x) = 2x4 + 7x3 + 14x2 + 11x − 10 as a product of its leading coefficient and itslinear factors. Hint: you can identify some zeros of P from its graph given below.

3. Determine the domain of the rational function

F (x) =x2 − 9x− 5

x2 + 14x + 48.

Write your answer in interval notation.

3. Consder the rational function F defined by

F (x) =x2 − 13x + 36

x2 − 10x + 24

(a) Find the domain of F ; write your answer in interval notation.

(b) Find all x-intercepts of F , if there are any.

(c) Find the y-intercept of F , if there is one.

(d) Write equations of all vertical asymptotes of F , if there are any.

3. Find all horizontal asymptotes, or explain why there is no horizontal asymptote, for therational functions F , G and H defined below.

(a) F (x) =15x9 + 12x5 − 5

6− 5x5 + 10x9

(b) G(x) =15x9 + 12x5 − 5

10x10 − 5x5 + 6

(c) H(x) =−5 + 12x5 + 15x9

6− 5x5 + 10x8

3. Find all slant and horizontal asymptotes for the rational function

F (x) =8x3 − 4x2 − 4x + 6

2x2 + 5

3. Use the rational function F defined below to answer the following questions.

F (x) =(x + 1)(x− 2)

x + 2

(a) Find the domain of F . Express answer in interval notation.

(b) Find the x-intercept(s) of F , if there are any.

(c) Find the y-intercept(s) of F , if there are any.

(d) Find the horizontal asymptote(s) of F , if there are any.

(e) Find the slant asymptote(s) of F , if there are any.

(f) Find all vertical asymptotes of F , if there are any. For each vertical asymptote, determinethe behavior of F just to the right, and just to the left of the asymptote.

(g) Use the information from (a) through (f), along with plotting some additional points asnecessary to sketch the graph of F along with all of its asymptotes.

3. Let f(x) =x2 − 3x + 2

x2 − 7x + 10.

(a) Simplify f and find its domain.

(b) Find equations for the vertical asymptote(s) for the graph of f .

(c) Find the x- and y-intercepts of the graph of f .

(c) For each vertical asymptote found in part (b), determine the behavior of f just to the rightand just to the left of the vertical asymptote.

(d) Find all values of c for which there is a “hole” in the graph of f above x = c.

(e) Find all horizontal asymptote of f .

(f) Use the information above and plot additional points a necessary to graph f .

3. Consider the rational function f(x) =3x2 − 9x + 6

x− 2.

(a) Does f have a vertical asymptote at x = 2?

(b) Sketch the graph of f .

Chapter 9, 10 and 11 Problems

5. A metallurgist made two purchases. The first purchase, which cost $652, included 19kilograms of an iron alloy and 21 kilograms of a lead alloy. The second purchase, at the sameprices, cost $952 and included 38 kilograms of the iron alloy and 26 kilograms of the lead alloy.Solve a system of linear equations to determine the price per kilogram of each alloy.

5. A goldsmith has two gold alloys. The first alloy is 35% gold; the second alloy is 63% gold.Set-up and solve a system of two linear equations to determine how many grams of each alloyshould be mixed to produce 70 grams of an alloy that is 41% gold?

5. Flying with the wind, a plane traveled a 1512 miles in 6 hours. Against the wind, the returntrip covering the same distance took 7 hours. Find the rate of the plane in calm air and findthe rate of the wind.

5. An investment adviser invested $10600 in two accounts. One investment earned 3% annualsimple interest, and the other investment earned 5% annual simple interest. The amount ofinterest earned for one year was $364. Set-up and solve a system of equations to determinehow much money was invested in each account?

5. Use algebra to solve the following system of equations.3x − 8y + 3z = −5−x + 3y − 4z = 5−4x + 3y + 3z = −8

5. Solve the following systems of equations.

(a)

{x − 3y + 6z + 5w = 2

y − 4w = 1

(b)

x − 5y + 2z + 3w = 6

−2x + 5y − 2z − 4w = −3x + w = 4

5. Consider the following system of equations.x + 4y − 2z = −4

y + 5z = 4(k − 3)z = −15

(a) Find the values of k for which (if possible) the system: (i) one solution, (ii) no solution,(iii) infinitely many solutions.

(b) Solve the system, if possible, when k = 0.

(c) Solve the system, if possible, when k = 6.

5. Consider the following system of equations.x + 2y − 3z = 4

y + 2z = −2(k2 − 13k + 36)z = k2 − 16

(a) Determine values of k for which this system has no solution, if possible.

(b) Determine values of k for this this system has infinitely many solutions, if possible.

(c) Determine values of k for this this system has a unique solution, if possible.

5. Find the equation of a parabola y = ax2 + bx + c that passes through the points (−3, 36),(−1, 14) and (2, 11). Then check that your equation works.

5. A coin bank contains only nickels, dimes and quarters. There are 8 fewer dimes than 2-timesthe number of nickels. The are 7 more quarters than 3-times the number of dimes. If the coinbank has a total of $15.95 in it, how many of each type of coin does it contain?

5. Use Gaussian elimination to solve the following system of equations−2x − 3y − 6z = −2

3x + 5y + 7z = 0−x − 1y − 4z = 1

5. Use Gaussian elimination to solve the following system of equations−2x − 5y − 7z = 1

3x + 9y + 9z = 4−x − 2y − 4z = 1

5. Find all values of A for which the following system has no solution.−4x − 5y − 14z = −1

4x + 7y + 10z = A−x − 1y − 4z = −1


5x + 13y + 14z = 3−x − 2y − 4z = 0


4x + 10y + 7z = −1−x − 2y − 3z = −1

− 4y + 8z = 0

5. Let A =

[4 2 −10 3 1

]and B =

[3 −4 3−1 −1 −3

].

(a) Find 3A

(b) Find B − 3A.

(c) Find 3A− 4B.

5. Let the matrices A and B be defined by

A =

[−1 −1

2 1

]and B =

[−1 0 3

1 3 −1

]

(a) Find the product AB if it exists, or explain why the product doesn’t exist.

(b) Find the product BA if it exists, or explain why the product doesn’t exist.


A =

2 1−1 −1

2 1

and B =

[−3 −2 −2

3 3 2

]

(a) Find the product AB if it exists, or explain why the product doesn’t exist.

(b) Find the product BA if it exists, or explain why the product doesn’t exist.


A =

−1 2 −33 −3 −3−1 −1 1

and B =

0 −1 −3−1 2 0

2 −2 2

(a) Find the product AB.

(b) Find A2.

5. Let E =

[4 −1 3−3 2 0

] 2 0 2 −1 −3 1 03 −1 2 −2 0 −1 1−2 1 −1 2 −3 0 −3

.

(a) What are the dimensions of E?

(b) Find the entry e24 in the product. (Don’t find the whole product, and write DNE if theentry does not exist)

5. (a) Find the following matrix product 7 −5 −21 2 04 −4 3

xyz

(b) Write the following matrix equation as an equivalent system of equations 7 −5 −2

1 2 04 −4 3

xyz

=

3−2−5

(c) Write the system

5x + 5y − 5z = −75x + 2z = 5−3x + 7y − 3z = 2

in matrix AX = B form.

5. Find the inverse of the matrix

−2 5 −12 −4 31 −2 2

, if it exists. Show all steps.

5. Find the inverse of the matrix A given below, if it exists. Show all steps.

A =

−4 −13 −105 17 11−1 −3 −3

5. Solve the system

2x − 4y + 7z = 3x − 2y + 3z = 0

−3x + 5y − 7z = −2using the fact that the inverse of 2 −4 7

1 −2 3−3 5 −7

is

1 −7 −22 −7 −11 −2 0

. (You must use the requested method).

5. Find the first three terms and 10th term of a sequence whose nth term is given by

an =(−1)n−1

2− 10n

5. Find the first three terms and 5th term of a sequence whose nth term is given by

an =(−1)n+1

n2 + 2

5. Find the 33th term of the sequence whose nth term is defined by

an =(n + 2)!

(n− 1)!, n ≥ 1

5. Evaluate the following sum10∑n=7

(−1)n(4n)

5. (a) Write an expression for an, the nth term of the sequence of even numbers2, 4, 6, 8, 10, 12 . . .

(b) Write an expression for bn, the nth term of the sequence of odd numbers 1, 3, 5, 7, 9, 11, . . .

(c) Write an expression for cn, the nth term of the sequence whose first six terms are

11, 13, 15, 17, 19, 21, . . .

(d) Write an expression for dn, the nth term of the sequence whose first six terms are

4, − 6, 8, − 10, 12, − 14, . . .

(e) Write an expression for en, the nth term of the sequence whose terms are multiples of 5,so its first six terms are

5, 10, 15, 20, 25, 30, . . .

(f) Write an expression for fn, the nth term of the sequence whose first six terms are

− 1, 4, 9, 14, 19, 24, . . .

5. Express the following sum in summation notation.

6

36+

7

49+

8

64+

9

81+

10

100

5. Expand the binomial (4v5 − 2w)5. (Show all steps; if you use Pascal’s triangle, write it asfar as you use it).

5. Use a binomial expansion to simplify the complex number

(2− 4i)4

and write the final answer as a complex number in standard form.

5. Find the seventh term of the binomial expansion of (3x− y3)11

5. Find the term of the binomial expansion (x + y)8 that contains x3y5

5. Find the term of the binomial expansion (x2 − 3y)11 that contains x14

Chapter 4 Problems

4. The graph of a function f is given below. On same graph sketch the inverse function of f ;notice that f goes through the points (0,−3) and (1,−2).

−6 −4 −2 2 4 6

−6

−4

−2

2

4

6

f

x

y

4. A function f that is one-to-one on its domain is graphed below.

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

x

y

For your reference, the points plotted on the graph of f are

(4,−2), (3,−1), (0, 0), (−5, 1)

(a) Sketch the graph of f−1 on the same graph.

(b) Find (i) f−1(1) and (ii) f−1(−2).

(c) Find (i) the domain of f−1 and (ii) the range of f−1.

4. Suppose f(x) = 3x5 + 5x + 1.

(a) Evaluate: f(−1), f(0), f(1), f(2).

(b) Given that f has an inverse function, use your answers from (a) to find f−1(107), f−1(9),f−1(1) and f−1(−7).

4. Find the inverse function of f(x) = 6x3 − 7. Once you have found f−1(x), verify thatf(f−1(x)) = x for all x.

4. (a) Find the inverse function of f(x) =3x + 2

8x− 6, and find its domain.

(b) Find the range of f .

(c) Solve the equation3x + 2

8x− 6= 3, if possible. Note. This is just asking you to solve f(x) = 3

for x.

(d) Solve the equation3x + 2

8x− 6=

3

8, if possible.

4. Let f(x) =x

2x + 4.

(a) Find f−1(x).

(b) Find the domain of f .

(c) Find the domain of f−1.

(d) Find the range of f .

(e) Find the range of f−1.

4. Use properties of exponents to solve 52x−9 = (58)x+8

(calculators should not be used).

4. Consider the function f(x) = 5x

(a) Complete the following table of values for f .

x −3 −2 −1 0 1 2 3

y

(b) Sketch a graph of f , and on the same coordinate axes sketch y = x and the graph of f−1(x)

(c) Using your answer from (b), sketch g(x) = 5x+3 − 6.

4. The graph of an exponential function y = bx is given on the graph below.

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

x

y

(a) Use the graph to estimate b.

(b) Using the graph from (a), graph y = b|x|.

(c) Using your graph from (b), graph y = b|x+3| − 6.

4. The graph of an exponential function y = bx is given on the graph below.

−6 −4 −2 2 4 6

−12

−8

−4

4

8

12

x

y

(a) Use the graph to estimate b.

(b) On the same graph, graph y = 2− bx.

4. The number of bass in a lake is given by

P (t) =4410

1 + 8e−0.08t

where t is the number of months that have passed since the lake was stocked with bass.

(a) How many bass were in the lake immediately after it was stocked?

(b) How many bass were in the lake 2 years after it was stocked? Round your answer to thenearest whole number.

(c) What will happen to the bass population as t increases without bound?

4. The population of a small city is currently 77000 and is growing at 5 percent per year. Thusthe population is given by

P (t) = 77000(1.05)t

where t is time measured in years from the present.

(a) What will the population of the city be in one year?

(b) According to this model, what will the population of the city be in 14 years from now?Express answer to the nearest whole number.

(c) Suppose Charles has an investment account that is growing a a rate of 5 percent per year,and he currently has 77000 dollars in the account. How much money will be in the account 14years from now? Express answer to the nearest dollar.

4. Exponential functions are ideal for dealing with large numbers (or extremely small numbers),because once we know the base, only the exponent changes. For example 106781.2346 and10781.8457 are numbers larger than many calculators will accept, yet properties of exponentsmake them easy to multiply (or divide). Indeed,

106781.2346 × 10781.8457 = 106781.2346+781.8457 = 107563.0803

Properties of exponents also make it easy to write the above answer in scientific notation

107563.0803 = 107563+.0803 = 100.0803 × 107563 ≈ 1.2031× 107563

Use properties of exponents to answer the following questions.

(a) Write 101798.186 in scientific notation. Use 5 significant figures in your final answer.

(b) Find the product 101798.186 · 1029.626 as a power of 10.

(c) Convert your answer in (b) to scientific notation. Use 5 significant digits in your answer.

4. (a) Evaluate log2 4. (b) Evaluate log2

(1

64

)(c) Evaluate log 1

24

4. Find the domain of the logarithmic function f(x) = log5(x2 − 16x + 63).

4. (a) Change the equation log 15

25 = −2 to exponential form.

(b) Change the exponential equation 64 = 1296 to logarithmic form.

4. Consider the functions f(x) = 5x and g(x) = log5 x

(a) Complete the following table of values for f(x) = 5x.

x −3 −2 −1 0 1 2 3

5x

(b) Complete the following table of values for g(x) = log5 x by filling in the x-values for thegiven y-values.

x

log5 x −3 −2 −1 0 1 2 3

What do you notice about the table of (a) and (b)?

(c) Sketch the graphs of f(x) = 5x and g(x) = log5 x on the same coordinate axes.

(d) Using your answer from (b), sketch h(x) = log5(x + 2)− 2.

4. Consider the functions f(x) =(16

)xand g(x) = log 1

6x

(a) Complete the following table of values for f(x) =(16

)x.

x −3 −2 −1 0 1 2 3

6x

(b) Complete the following table of values for g(x) = log 16x by filling in the x-values for the

given y-values.

x

log6 x −3 −2 −1 0 1 2 3

What do you notice about the table of (a) and (b)?

(c) Sketch the graphs of f(x) =(16

)xand g(x) = log 1

6x on the same coordinate axes.

(d) Using your answer from (b), sketch h(x) = log 16(x− 1) + 4.

4. Consider the function f(x) = log 16(x + 6) + 2

(a) Determine the domain of f . Write your answer in interval notation.

(b) Solve the equation y = log 16(x + 6) + 2 for x by converting the equation to exponential

form.

(c) Use your answer in (b) to complete the following table of values where y has been chosenfirst.

x

y 5 4 3 2 1 0 −1

(d) With the the help of your table in (c), sketch the graph of f(x).

4. The common log of the positive numbers M and N are given below.

log(M) = 3935.851 and log(N) = −3935.851

(a) Express M and N as powers of 10.

(b) Express M and N in scientific notation. Use 6 significant figures in your answers.

4. Use properties of logarithms to fully expand log 110

(100y7√x11z11

)in terms of log 1

10x, log 1

10y,

log 110z and number(s); simplify your answer (assume x > 0, y > 0, and z > 0).

4. Use properties of logarithms to write 2 + 10 log9 y− 52

log9 x− 72

log9 z as a single logarithmwith coefficient 1.

4. Use properties of logs and exponents to evaluate: (a) e4 ln 3 (b) 23 log2 4 (c) log4(2568)

4. (Richter Scale) The magnitude of an earthquake of intensity I on the Richter scale is

M = log

(I

I0

)where I0 is the intensity of a zero-level earthquake.

(a) Find the magnitude to the nearest 0.1 of an earthquake that has an intensity of I = 31623I0.

(b) Write the formula M = log

(I

I0

)in exponential form, and then solve for I.

(c) Use the formula in (b) to find the intensity of an earthquake that measures 7.0 on theRichter scale.

(d) How many times more intense is an earthquake with a magnitude of 6.5 than an earthquakewith a magnitude of 5.3? Express answer to nearest whole number. (To do this, find theintensity of each earthquake using your formula from (b), and then divide to find the ratio ofthe intensities).

4. The range of sound intensities that the human ear can detect is very large, so a logarithmicscale is used to measure them. The decibel level (dB) of a sound is given by

dB(I) = 10 logI

I0

where I0 is the intensity of sound that is barely audible to the huma ear.

(a) How many times as great is the intensity of a sound that measures 151 decibels whencompared to a sound that measures 91 decibels?

(b) Find the decibel level of a sound whose intensity is 61 times as intense as a sound thatmeasures 91 decibels. Round your answer to one decimal place.

4. The percentage of a certain radiation that can penetrate x millimeters of lead shielding isgiven by

P (x) = 100e−1.5x

(a) Find the percentage of radiation that can penetrate 7 mm of lead shielding. Use at least 3significant figures in your answer.

(b) How many millimeters of lead shielding are required so that less than 0.003 percent ofradiation will penetrate the shielding? Round answer to one decimal place (nearest tenth of amillimeter).

4. Which is bigger 515516 or 516515?

4. Logarithms are ideally suited for dealing with large numbers. Use properties of logarithmsand exponents to answer the following questions.

(a) Use properties of logarithms to write 5521 in scientific notation. Use 4 significant figures inyour final answer.

(b) Find the product 5521×3321717. Express your answer in scientific notation. Use 4 significantfigures in your answer.

(c) Find the quotient 3321717÷5521. Express your answer in scientific notation. Use 4 significantfigures in your answer.

4. The common logs of the positive numbers M and N are given below.

log(M) = 3639.394 and log(N) = 293.949

(a) Find log(MN), log(M/N) and log(N/M).

(b) Find the product MN as a power of 10.

(c) Write the product MN in scientific notation. Use 6 significant figures in your answer.

(d) Find the quotient M/N , and write your answer in scientific notation. Use 6 significantfigures in your answer.

(e) Find the quotient N/M , and write your answer in scientific notation. Use 6 significantfigures in your answer.

4. (a) Find the exact solution to 2(6x)− 11(36x) = 0.

(b) Use a calculator to express your answer in (a) to five decimal places.

4. Solve the equation 24x−4 = 6−4x−2 for x. Leave answer in exact form.

4. (a) Find the exact solution to the equationex − 0.7e−x

ex + 5e−x=

1

6. You don’t have to simplify

the exact solution.

(b) Use a calculator to express your answer in (a) to six decimal places.

4. Find the exact solution(s) to the equation ex +42e−x = 13. Verify that your solutions work.

4. Solve the equation log4(x2 − 6x + 9) = 1.

4. Always be careful to check that your solutions work when solving logarithmic equations,because logarithms and their properties are only defined for positive numbers. See whathappens when you solve the equation:

ln(x) + ln(x− 7) = ln(6x− 42)

For this, combine the logs on the left side using: ln(M) + ln(N) = ln(M + N) when M > 0and N > 0, and then use the property ln(E) = ln(F ) implies E = F when ln(E) and ln(F )are defined.

4. The population of a city is currently 435000 and is expected to grow at a rate of 9.5 percentper year for the foreseeable future. Its population is given by P (t) = 435000(1.095t) where t isthe number of years from today.

(a) What will the population be in 10 years? (Express answer as a whole number)

(b) At this rate of growth, how long (in years) will it take the population to double? How long(in years) would it take the population to quadruple? Express answers to 1 decimal place.

(c) If this growth rate could continue, how long (in years) would it take for the population toreach 3,000,000 people? Express answer to 1 decimal place.

4. (Interest Income) Use the properties of logarithms and exponentials, along with the com-pound interest formula

A = P(

1 +r

n

)ntto answer the following questions.

(a) Suppose $16000 is invested at an annual interest rate of 3.5% compounded monthly. Howmuch will it be worth after 14 years? Express answer to nearest penny.

(b) How long will it take until the investment is worth $330000? Express your answer in yearsrounded to one decimal place.

4. (Exponential Growth) The population of bacteria in a vat of potato salad at Bob’s All DayBuffet is modeled by P (t) = P0e

kt. At noon there were 1500 bacteria present and at 1:00 pmthere were 2250 bacteria present.

(a) Find the specific model for P (t) (i.e., use the information given to find P0 and k, and plugthose values into P (t) = P0e

kt), and then use it to answer the following questions.

(b) How may bacteria were present in the potato salad at 11:30 am when was placed in thebuffet? Express answer to nearest whole number.

(c) How may bacteria were present in the potato salad at 3:00 pm when the potato salad wasremoved from the buffet?

(d) If the potato salad had been allowed to remain in the buffet indefinitely, and the modelfor the bacteria remained valid, how many hours after noon, when there were 1500 bacteriapresent would it have taken for the bacteria population to reach 32625. Express answer inhours, rounded to nearest decimal place.

(For all of these questions, assume no one ate took any potato salad because of its funny smelland hence the population of bacteria remained in tact and followed the given growth model).

4. (Carbon Dating) Carbon-14 has a half-life of 5730 years, and satisfies the exponential-decay

equation N(t) = N0

(1

2

)t/5730

.

(a) If an ancient scroll is discovered to have 59.5% of its original Carbon-14, how old is thescroll? Round answer to nearest year.

(b) What percentage of a bone’s original Carbon-14 would you expect to find remaining in abone that is 4400 years old? Round to the nearest tenth of one percent.

4. An unknown radioactive element decays into non-radioactive substances. In 690 days theradioactivity of a sample decreases by 76 percent, that is 24 percent of the original substanceremains.

(a) What is the half-life of the element?

(b) How long will it take for a sample of 100 mg to decay to 90 mg?

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