5. Perform the indicated operation and simplify each of the following expressions:

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1 Precalculus Worksheet.5 1. What is - 1? Just because we refer to solutions as imaginar does not mean that the solutions are meaningless. Fields such as quantum mechanics and electromagnetism depend on the mathematics of imaginar numbers. When engineers design airplane wings or cell-phone towers, imaginar numbers are vital to their calculations. Applications abound in electrical engineering, vibration engineering, polmer 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 MP plaers. More heav dut applications include the design of missile guidance sstems. While we are not read to dive into these topics et, we are read to learn to simplif the square root of negative numbers correctl.. Evaluate the following epressions: (no calculator) a) - 5 b) - 60 c) (-6) -4( )( - ) d) 5-4(-6)(- ) When adding, subtracting and multipling comple numbers, pa close attention to whether ou are adding, subtracting, or multipling. MOST people that miss these questions do so because the decide to multipl everthing. 4. What is i? 5. Perform the indicated operation and simplif each of the following epressions: a) ( 7- i) + ( 6- i) b) ( 7-i) -( 6- i) b) ( 7-i)( 6- i) d) ( 5i- )( i+ 1) 6. Find the conjugate of each comple number then multipl them together. a) 7+ 5i b) -5- i When ou multipl a comple number b its conjugate the answer is a REAL number.

2 In order to divide comple numbers, ou will multipl the numerator and denominator b the conjugate of the denominator so that the denominator will be a real number. : If ou came directl from regular Algebra, ou ma never have seen this please do not hesitate to ask for some help! Eample: 5- i ( 5-i) ( 7-5i) 7+ 5i ( 7+ 5i) ( 7-5i) 5-5i- 14i+ 10i 49-5i + 5i - 5i 5-9i i i Divide the following comple numbers. Write our answer in the form a + bi. a) + 8i -5-i b) + i 9-4i 9. Solve the following quadratic equations b using the quadratic formula. a) b) - 1-5

3 Solving quadratic equations can also be done b using completing the square. Completing the square is most useful when the coefficient of is equal to 1 (and the coefficient of is even). Go back to Section P5 or.1 if ou need a refresher on how to complete the square. 10. Solve the following equation b completing the square and compare it to question 9a above: The impedance of an electrical current is a wa of measuring how much the circuit impedes the flow of electricit. The impedance can be a comple number. A circuit is being designed that must have an impedance that satisfies the f , where is a measure of the impedance. Find the zeros of the function. (Use an function ( ) method ou like). 1. Draw a picture of (or eplain wh ou are not able to draw) each of the following: a) a quadratic function having onl one real number root. b) a quadratic function having onl one comple root. c) a quadratic function with two real roots. d) a quadratic function with two comple roots.

4 1. Previousl, we used snthetic division to show that a value of was indeed a zero of a function. If ou were told that one of the solutions to the equation is + i, use snthetic division to show that it reall is a zero. Using the result of our snthetic division, what is the other solution? : You will use snthetic division with comple numbers in lesson.6 In the last eample the conjugates + i and i were both solutions. This is not a coincidence. In a polnomial if a + bi is a zero, then the conjugate a bi is also a zero. Using the last eample, this means that the epression could be factored to é -( + i) ùé-( -i) ù ë ûë û. 14. Write the polnomial function of minimum degree in standard form whose given zeros and their multiplicities include those listed below. a) 4i b) 5 and 4i c) (multiplicit ), and another zero of + i OPTIONAL EXTRA PRACTICE: For more practice on each of the following topics refer to pages 4-5 of our book. Given zeros of a polnomial, find the equation of the polnomial #5 15

5 Pre Calculus Worksheet.6 1. State the Fundamental Theorem of Algebra in our own words. Wh do we need it?. Use the equations and the graphs below to do the following: i) Identif the number of comple zeros each equation has according to the Fundamental Theorem of Algebra. ii) Determine how man real and non-real zeros each function has according to the graph. a) 4 f( ) 7 6 b) f ( ). How man real zeros are ou guaranteed to have if ou have an odd degree polnomial? 4. Is it possible to find a polnomial f () that has the degree 4 with the given zeros? Eplain wh or wh not. a), 1 + i and 1 i b) 1 i and 4 + i 5. Draw a picture of (or eplain wh ou are not able to draw) each of the following: a) a quartic function having onl one comple root. b) a quartic function with two real roots.

6 [No Calculator] Using the given zero, find all comple zeros of the polnomial function f( ) 5 ; zeros: 1 and 4 7. h ( ) ; zero: + i 5 8. Use factoring to find all the zeros of f( ) Use the graph below to help factor g ( ) The scale on the -ais is 1 unit / tick. 4

7 Use our graphing calculator to find and verif 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 es, now!!!!!! 10. f ( ) f 4 ( ) 4 1. Find the unique polnomial function of degree 4 with zeros, 1, and i where f(0) 0. Write our 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 4-5 of our book. (If ou need one, just ask) Find zeros of a polnomial given comple zero #7 Factor a polnomial into linear factors and/or irreducible quadratic factors #7-40

8 Pre Calculus Worksheet.7 Da 1 No Calculator should be used on this worksheet. Match the function with the corresponding graph b considering end behavior and asmptotes f ( ). f ( ) f ( ) f ( ) For each function find the following (if the eist). a. End Behavior including the equations of horizontal or slant asmptotes b. Vertical Asmptote(s). Distinguish between VA and Holes f ( ) 6. f ( ) 7. f ( ) f ( ) f ( ) 10. f ( )

9 When a rational function has a linear polnomial divided b a linear polnomial, we have a special ration function that makes the 1 Inverse Linear parent function f ( ). We use long division rewrite a rational function into the Inverse Linear form. Consider the function k( ) we divide to get or k( ) So, k() is simpl f ( ) 4 1 with a vertical stretch of 4, translated right 1 and up. We can now easil graph the function. 11. After reading the information above, graph k (). For 1 15, use long division to obtain a function whose parent is sure to list the HA and VA. f ( ) 1. Describe the transformation and graph the function. Be 1. 1 f ( ) 14. g( ) h ( ) 4

10 Pre Calculus Worksheet.7 Da For each function find the following (if the eist). a. End Behavior including the equations of horizontal or slant asmptotes Write as an equation of a line. b. i. Vertical Asmptote(s) Write as an equation of a line. ii. Hole(s) Write as an ordered pair. d. - intercept(s) Write as an ordered pair. e. -intercept Write as an ordered pair. f. Graph without a calculator. Identif enough additional points in each region to determine shape of graph f ( ). f ( ) f ( )

11 4. f ( ) f ( ) f ( ) f ( )

12 Precalculus Worksheet.9 1. Use the graph of g () at the right to answer each question. a) When is g () > 0? b) When is g () > 0? c) When is g () < 0? d) When is g () < 0?. Solve the following inequalit algebraicall: ( ) ( )( ) < 0.. Compare our answer from question to question 1c. What do ou notice? 4. Solve the following inequalit using a graph AND algebraicall: ( )( ) 5- + ³ 0 5. Consider the function f ( ) a) When does f () 0? ( - 4) ( 7)( 5) + -. b) When is f () undefined? c) Create a sign chart to determine when f () > 0. d) Use the sign chart ou created in part c to determine when f () < 0.

13 5. Solve each inequalit b first completing factoring the left side and creating a sign chart. a) > 0 b) < A sign chart in question 5b involved a rational function that was less than zero. You ma first need to add or subtract fractions in order to create a rational function. You ma 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 - > b) ³ +

14 In lesson.7 we graphed rational functions b first finding the ke attributes of rational functions Find the ke attributes (all asmptotes, intercepts, and holes) and create a graph to solve the inequalit: Using what ou know about sign charts, solve the inequalit above. Could making a sign chart help ou create graphs of rational functions? Eplain how. Need more practice? Ask our teacher to check out a tetbook. Like # and #: Alread Factored Polnomial Inequalities: page 65 #7 and 8 Like #5a and #5b: Factoring Required Polnomial Inequalities: page 65 #9 0 Like #4 and #5c: Rational Inequalities (with or without factoring): page 66 #5, 6 40 Like #6 and #7: Write the Rational Inequalit as a Single Fraction First: page 66 #47, 49