Pre Calculus Worksheet P.1 1. Describe in words and graph each interval of real numbers. a) x < 2 b) –2 < x < 5 c) , 7 d) [–3, 3] 2. Graph the interval described in words and use interval notation to describe the interval. a) x is negative b) x is greater than or equal to 2 and less than or equal to 6 3. Explain why each of the following should NEVER be written: a) 5 < x > 7 b) 3 > x > 8 4. Describe the graphs below using both inequalities and interval notation. Later in the course we will be using sign charts. Sign charts are simply number lines with “pos” or “neg” signs on them to represent whether or not the function or equation is positive or negative on that interval. Suppose the sign chart below represents when the function f (x) is positive “pos” or negative “neg”. Use the sign chart to answer the questions 5 and 6 in interval notation. 5. When is f (x) < 0 ? … (i.e. when is the function negative?) 6. If f (x) = 0 when x = –2 and x = 1, but not when x = 7, when is f (x) > 0 ? 7. Use both inequality AND interval notation to describe the set of numbers. Define any variables used.

a) In Excel, you can create an if-then statement that will enter a specific value or text into a box depending on given conditions. Suppose you would like to enter a value of “YES” into cell A8 if the value in cell A7 is between 4 and 16 including both.

b) A programmer wants to create an alertbox if the user inputs a value that is at least 150.

3 6 9 12 15 18 5 10 15 20 25 30

7

pos+

1 –2

pos pos neg f (x)

8. Identify the base in each of the following expressions, then evaluate each expression. a) 24− b) ( )24− c) ( )33− d) 33− e) 22 1x + f) ( )22 1x + 9. Match the following equations with the property illustrated. Each letter is only used once. 3 3x y xy 2 2a b ba 2 2 0a b a b 2 23 0 3x x a x y ax ay

12 12

xx

1 x y x y a b c c a b 2 3 2 3x y z a x y z a

A) Commutative Property of Addition B) Commutative Property of Multiplication C) Associative Property of Addition D) Associative Property of Multiplication E) Identity Property of Addition F) Identity Property of Multiplication G) Inverse Property of Addition H) Inverse Property of Multiplication I) Distributive Property

10. Simplify each expression. a) ( )3 24 2xy x ax+

b) ( )3 2 5a a b+

Many times we need to undo the distributive property. This is the beginning of factoring. When you undo the distributive property, we say you have factored out the greatest common factor. 11. Factor out the GCF from each expression. a) 23 6x xy− − b) 3 2108 36 60x x x− + c) 236 + 52 12. Describe in your own words what the phrase “Take out a GCF” means.

Pre Calculus Worksheet P.3 1. Solve each of the following equations algebraically: a) 4 2 5y y b) 71

2 8x c) 1 12 3 1x

d) 2 3

4 5 3x x e) 5 2 18 2 3

t t 2. The formula for the perimeter of a rectangle is given by P = 2L + 2W. Solve this equation for W. 3. The formula for the area of a trapezoid is given by ( )1

1 22A h b b= + . Solve this equation for b1. 4. A formula relating Fahrenheit to Celsius is ( )5

9 32C F= − . Solve this equation for F. 5. A satellite orbiting the planet has a mass m, a velocity v, and a radius from the center of the planet equal to R. The

centrifugal Force F acting on that satellite is given by the equation 2mvF

R . Solve this equation for R.

6. You have scores of 68, 82, 87, and 89 on your first four tests, and you have one more to take. In order to get a B you must have an average between 79.5 and 89.5. What is the lowest score you could get on your last test in order to get a B. 7. When you solve an inequality, when does the inequality change direction. Explain why. 8. Solve each of the following inequalities and write your answer in interval notation. a) 2 6 9x b) 1 3 2 7x c) 5 7

4 3x d) 2 534 2y

e) 3 4 2 3

6 8 2y y y f) 1 12 33 2 4 3x x x

Pre Calculus Worksheet Factoring 1. Learning to recognize the difference of two perfect squares is more difficult than factoring them. Factor the following expressions, and write something on this paper that will help you recognize this in the future. a) 2 25a − b) 216 n− 2. Factor the following quadratic expressions completely: a) 2 8 15b b+ + b) 26 35 6d d+ − c) 210 19 6t t− + d) 3 24 5p p p− − e) 3 212 14 40f f f− − f) 4 3 220 23 6e e e− + 3. Factor the following equations completely using the grouping method: a) 6 6ax ay x y+ + + b) 2 27 3 21mp m p+ + + c) 3 27 9 63b b b 4. Factor each of the following expressions:

a) 26 5 3 11 5 3 7x x b) 25 8 9 47 8 9 18x x

Pre Calculus Worksheet P.5 The main purpose of section P.5 is for you to be able to solve quadratic functions in the most efficient (quickest) method possible. In order to do this you should learn as many different methods as possible. 1. Graphically solve the equation 3 2 2 3 0x x x+ + − = . Describe your method. 2. Graphically solve the equation 2 4 4x x . Describe your method. 3. Solve the equation 22x x= graphically two different ways … keep in mind you should get the same answer(s). a) Solve by finding the intersections. b) Solve by finding the zeros. 4. The table below shows the x- and y-coordinates of the equation 2 2 1y x x . State the expression and give the zero as accurately as can be read from the table. 5. Solve each of the following equations by using factoring: a) 3 11 20x x b) 3 2 9 9 0x x x

x y 0.4 -0.04 0.41 -0.0119 0.42 0.0164 0.43 0.0449 0.44 0.0736 0.45 0.1025 0.46 0.1316

6. Solve each of the following equations by “extracting square roots”. a) 23 5 43x b) 23 49

4 322 x 7. Solve each of the following equations by “completing the square” a) 2 6 7x x b) 23 2 0x x 8. Solve the equation using the quadratic formula: 5 12x x Keep in mind that for the purposes of quizzes and tests, there will be times that you will be asked to use a specific method. However, in general, you may use whatever method works for you and is appropriate. The second portion of this section is for you to solve equations involving absolute value.

9. Solve each of the following equations graphically AND algebraically: a) 8 2t b) 5 2 3 1 9x c) 2 4 1 4x x 10. The following equations represent things you should be able to do to this point. Solve each algebraically:

a) 2 5 3 8 4 5x x b) 23 5 17x c) 3 14 3 12x x

d) 3 1 5x e) 2 3 4x x x f) 24 6x x

Pre Calculus

Worksheet P.6

For questions 1 – 4, solve the inequality algebraically. Write your answer in interval notation.

1. 8 17x 2. 3 2 4x

3. 3 2 9 9x 4. 5 2 3 8 13x

5. The 16 4 , but the equation 2 16x has two solutions, 4x . The reason for this is the fact that

2x x .

Using this fact, solve the following inequalities algebraically:

a) 2 9 0x b)

22 3 5 13x

6. Solve the following inequalities graphically. Write your answer in interval notation.

a) 22 17 21 0x x b)

32 2 5x x

c) 22 5 3 8x x d)

3 0x x

7. For a certain gas, P = 400/V, where P is the pressure and V is the volume. If 20 < V < 40, what is the corresponding

range of P ?

8. An open-top box is to be made by cutting congruent squares of side length x from the corners of a 20- by 25-inch sheet

of tin and bending up the sides (see figure below).

a) Write an equation for the volume V as a function of x.

What is the domain of this function?

b) What size corner squares should be cut to yield a box with a volume of 300 in3?

c) What size corner squares should be cut to yield a box with a volume more than 300 in3?

d) What size corner squares should be cut to yield a box with a volume of at most 300 in3?

e) How large should the square be to make the box hold as much as possible? What is the resulting volume?

x

x

x

x x

x

x

x

25

"

20"

Pre Calculus Worksheet 2.5 1. What is 1 ? Just because we refer to solutions as imaginary does not mean that the solutions are meaningless. Fields such as quantum mechanics and electromagnetism depend on the mathematics of imaginary numbers. When engineers design airplane wings or cell-phone towers, imaginary numbers are vital to their calculations. Applications abound in electrical engineering, vibration engineering, polymer science, navigation, and more. Applications abound in the real world, touching our lives via design of popular features such as the vibrating ringer in our cell phones or the bass boosters in our MP3 players. More heavy duty applications include the design of missile guidance systems. While we are not ready to dive into these topics yet, we are ready to learn to simplify the square root of negative numbers correctly. 2. Evaluate the following expressions: (no calculator) a) 25 b) 60 c) 25 4 6 3 When adding, subtracting and multiplying complex numbers, pay close attention to whether you are adding, subtracting, or multiplying. MOST people that miss these questions do so because they decide to multiply everything. 4. What is 2i ? 5. Perform the indicated operation and simplify each of the following expressions: a) 7 3 6i i b) 7 3 6i i c) 7 3 6i i 6. Find the conjugate of each complex number then multiply them together. What do you notice about your answers? a) 4 9i b) 5 3i 8. Divide the following complex numbers. Write your answer in the form a + bi.

a) 2 85 3

ii

b) 2 39 4

ii

9. Solve the following quadratic equation by using the quadratic formula. Write your answers in a + bi form where applicable. 2 11 5 8x x x Solving quadratic equations can also be done by using completing the square. Completing the square is most useful when the coefficient of x2 is equal to 1 (and the coefficient of x is even). … “Go back to Section P5 if you need a refresher on how to complete the square.” 10. Solve the following equation by completing the square and compare it to question 9a above: 2 11 5 8x x x 12. Draw a picture of (or explain why you are not able to draw) each of the following: a) a quadratic function having only one real number root. b) a quadratic function having only one complex root. c) a quadratic function with two real roots. d) a quadratic function with two complex roots. 13. Previously, we used synthetic division to show that a value of x was indeed a zero of a function. If you were told that one of the solutions to the equation 2 4 5 0x x is x = 2 + i, use synthetic division to show that it really is a zero. Using the result of your synthetic division, what is the other solution?

: You will use synthetic division with complex numbers in lesson 2.6

In the last example the conjugates x = 2 + i and x = 2 – i were both solutions. This is not a coincidence. In a polynomial if x = a + bi is a zero, then the conjugate x = a – bi is also a zero. Using the last example, this means that the expression

2 4 5x x could be factored to 2 2x i x i . 14. Write the polynomial function of minimum degree in standard form whose given zeros and their multiplicities include those listed below. a) x = 4i b) x = –5 and x = 4i c) x = –2 (multiplicity 2), and x = 2 + 3i 15. Given the complex numbers w = – 2 + 4i and v = 5– 2i, answer the questions below… a) Plots points in the complex plane. b) Find 5 2i− and explain what this means. c) Find the distance between w and v. d) Find the midpoint of the segment connecting w and v. OPTIONAL EXTRA PRACTICE: For more practice on each of the following topics…refer to pages 234-235 of your book. Given zeros of a polynomial, find the equation of the polynomial… #5–15

Real axis

Imaginary axis

Pre Calculus Worksheet 2.6 1. State the Fundamental Theorem of Algebra in your own words. Why do we need it? 2. Use the equations and the graphs below to do the following: i) Identify the number of complex zeros each equation has according to the Fundamental Theorem of Algebra. ii) Determine how many real and non-real zeros each function has according to the graph. a) 4 2( ) 2 7 6f x x x x= − − + b) 3 2( ) 3 3f x x x x= + − + 3. How many real zeros are you guaranteed to have if you have an odd degree polynomial? 4. Is it possible to find a polynomial f (x) that has the degree 4 with the given zeros? Explain why or why not. a) –3, 1 + 2i and 1 – i b) 1 – 3i and 4 + i 5. For question 4, if there was a possible polynomial with the given zeros, find that polynomial in standard form.

[No Calculator] Using the given zero, find all complex zeros of the polynomial function.

6. 4 3 2( ) 5 3 3 3 2f x x x x x= + + + − ; zeros: x = –1 and 2

5 7. 4 3 2( ) 4 4 36 45h x x x x x= − − + − ; zero: x = 2 + i

8. Use factoring to find all the zeros of 3 2( ) 3 4 12f x x x x= + + + 9. Use the graph below to help factor 4 3 2( ) 3 11 10 4 8g x x x x x= − + − + . The scale on the x-axis is 1 unit / tick.

Use your graphing calculator to find and verify all zeros. Then, write the function as a product of linear and irreducible quadratic factors. IF YOU DON’T KNOW WHAT “LINEAR AND IRREDUCIBLE QUADRATIC FACTORS” MEANS, please ask … yes, now!!!!!! 10. 3 2( ) 15f x x x x 11. 4 3 2( ) 3 3 3 4f x x x x x= + − + − 12. Find the unique polynomial function of degree 4 with zeros x = 3, x = –1, and x = 2 – i where f(0) = 30. Write your function as both the product of linear and irreducible factors, as well as, in standard form. OPTIONAL EXTRA PRACTICE: For more practice on each of the following topics…refer to pages 234-235 of your book. (If you need one, just ask) Find zeros of a polynomial given complex zero…#27–32 Factor a polynomial into linear factors and/or irreducible quadratic factors…#37-40

Pre Calculus Worksheet 2.8

1. 5 2 196 2 3 6

mm

2. 4 3cc

3. 2

2 4 83 3 9x x x

4. 2

2 3 2 23 4 12

xx x x x

5. 2

2 6 183 3

xx x x x

6. 2

3 3 31

xx xx x

7. 2

3 1 75 2 3 10

xx x x x

8. 2

10 7 8 82 8 2 816

yy yy

9. 2

1 5 2 5( 4) ( 1)3 4

xx x x xx x

10. Solve for x: 111

yx

11. The total electrical resistance R of two resistors connected in parallel with resistances R1 and R2 is given by

1 2

1 1 1R R R

One of the resistors has a resistance of 2.3 ohms. Let x be the resistance of the second resistor. a) Express the total resistance R as a function of x. b) Find the resistance of the second resistor if the total resistance of the pair is 1.7 ohms.

12. Consider all the rectangles with an area of 200 m2. Let x be the length of one side of such a rectangle. a) Express the perimeter P as a function of x. b) Find the dimensions of a rectangle whose perimeter is 70 m. c) Find the dimensions of the rectangle that has the least perimeter. What is the least perimeter? 13. Explain the difference between the following three problems. Are there any similarities in their solution?

a) Write the expression as a single quotient: 22 4x x

x x

b) Solve: 2 02 4

x xx x

c) Solve: 2 02 4

x xx x

Precalculus Worksheet 2.9 1. Use the graph of g (x) at the right to answer each question. a) When is g (x) > 0 ? b) When is g (x) > 0 ? c) When is g (x) < 0 ? d) When is g (x) < 0 ? 2. Solve the following inequality algebraically: 23 1 1 0x x x . 3. Compare your answer from question 2 to question 1c. What do you notice? 4. Solve the following inequality using a graph AND algebraically: 35 2 3 0x x

5. Consider the function

2 4

7 3 5x x

f xx x

.

a) When does f (x) = 0 ? b) When is f (x) undefined? c) Create a sign chart to determine when f (x) > 0 . d) Use the sign chart you created in part c to determine when f (x) < 0.

6. Solve each inequality by first completing factoring the left side and creating a sign chart.

a) 3 24 4 0x x x b) 2

2

3 10 06 9

x xx x

7. A sign chart in question 5b involved a rational function that was less than zero. You may first need to add or subtract fractions in order to create a rational function. You may also need to make one side equal 0 before using a sign chart to solve. Solve the following problems using a sign chart.

a) 1 2 02 1x x

b) 4 5 3

2x

x

Need more practice? Ask your teacher to check out a textboo if you don’t have one... Like #2 and #4: Already Factored Polynomial Inequalities: page 265 #7 and 8 Like #6a: Factoring Required Polynomial Inequalities: page 265 #9 – 20 Like #5 and #6b: Rational Inequalities (with or without factoring): page 266 #25, 26 – 40 Like #7: Write the Rational Inequality as a Single Fraction First: page 266 #47, 49

Pre Calculus Worksheet 3.5 Use the properties of logarithms to expand the expression. Simplify, if possible.

1. ( )3

5 63log 81x y⋅ 2. 1/3

5

6ln

xeh

In 3-4, condense the expression to write the logarithm of a single expression. Simplify, if possible. 3. ln18 2 ln 6 ln 4− + 4. log 5 3 log logb b bx y− − For 5-6, let log 3bx = and log 4by = . Expand the logarithm. Then, write each expression in terms of x and y.

5. 9

log4b

6. log 0.75b

7. Evaluate each logarithm with your calculator. Explain why your answer is reasonable. No, the reason is not because the calculator said so!

a) 3log 29 b) 0.4log 5.2 There are two algebraic ways to solve an exponential equation… 8. Apply a common logarithm to both sides of the exponential equation: 1.05 2x = . Use the properties of logarithms to solve for x. 9. Put the equation 1.05 2x = into logarithmic form. Solve for x with your calculator. What do you notice about your answers to questions 8 and 9?

There are two algebraic ways to solve a logarithmic equation… 10. Apply a base to both sides of the logarithmic equation to “undo” the logarithm: ln 3.5x = . Solve for x. 11. Put the equation ln 3.5x = into exponential form. Explain how this process is the same as that of the previous question and how it is different. What do you notice about your answers to questions 1 and 2? Solve each equation algebraically showing all work. Round final answers to the thousandths. Remember to check your answers!

12. 48 20(0.5)x

= 13. 19 5 72x+⋅ = 14. 431 6 43x

e− + = 15. 5log (1 2 ) 3x− = 16. ( )72 ln 6x = 17. ( )74 9 log 5 3 1x− − = 18. ( ) ( )log 5 log 1 2x x+ − = 19. 3 3log ( 2) log ( 3) 2a a+ − − = 20. 4 4log ( ) 2 log ( 10)x x− = − + Examine the following questions, noting how they are different from questions 18-20. Then, solve. 21. 3 3 3log 5 log log 2x− = 22. 5 5 53 log 2 log log (24 2 )x x+ = +