COMPLEX NUMBERS. Algebra 2 & Trigonometry

Transcription

1 COMPLEX NUMBERS Algebra & Trigonometry

2 Name: Topic Pages Day 1 Imaginary Numbers / Powers of i 3 4 Day Graphing and Operations with Complex Numbers 8 13 Day 3 Dividing Complex Numbers Day 4 Complex Roots of Quadratic Equations 18 3 Day 5 Nature of the Roots (1) 4 7 Day 6 Nature of the Roots () 8 3 Day 7 Review 33 35

3 Steps to Evaluate Radicals with a Negative Radicand Day 1 Imaginary Numbers / Powers of i An equation, such as x = 9, has no solution in the real number system. However, a solution does exist in the system of imaginary numbers. By definition, 1 is defined as i, the imaginary unit. Since 9 = 9 1, and 1 = i, we can simplify 9 as 3i. 1. Remove the negative sign from under the radical.. Place an i in front of the radical sign. 3. Simplify the radical. Directions: Simplify each number and express in terms of i. 1.) 5.) ) 18 4.) 5 1 Steps in the Calculator: MATH BUTTON Important Powers of i Remember: i = 1 NUM (Arrow over once) i 0 = 1 3: ipart (option #3) i 1 = i i = 1 i 3 = -i Directions: Write each power of i in simplest terms ) i 6.) i 15 7.) 44 i 8.) i 97 3

4 ) What is the sum of 7i and15i? 10.) What is the product of 0 4i and 6i 13? Practice Problems ) The sum of 6i and 13i is (1) -19i (3) -19 () 19i (4) ) Find the product of 10i and 7i. (1) -70i (3) -70 () 70i (4) ) Simplify: ( 10i ) (1) -100i (3) -100 () 100i (4) ) Which of the following is not equal to the other three? (1) i 19 7 (3) i 9 35 () i (4) i ) When 6i is multiplied by 8i, the result is (1) -48i (3) -48 () 48i (4) ) The expression 48 is equivalent to 4 (1) 3 3 (3) 3i 3 3 () 3i 3 (4) i 3 4

5 Day 1 Imaginary Numbers / Powers of i HOMEWORK **Complete any Practice Problems from class work that have not been completed** Directions: Simplify. 1.) 8.) a b 5 3.) x 4 y Ans: i 7 Ans: 10 Ans: ab i b 10 xyi y Directions: Perform the indicated operation and express in simplest form of i. 4.) ) ) ) Ans: 1i Ans: 3i 8.) ) Ans: 19i Ans: 7i 5 Directions: Write each power of i in simplest terms ) i 11.) i Ans: 0 1.) 14 i Ans: 13i Ans: i Ans: i Ans: 1 5

6 13.) i 3i 14.) 3i 40i ) i + 7i 5 7 Ans: 1i 0i 16.) 16 10i 3 Ans: 7 Review Ans: 5i 17.) Solve the system of equations algebraically: x 14 = y y 1= x Ans: i 6 Ans: (5, 11), (-3, -5)

7 18.) One of the students in class was absent the day the class learned the technique of completing the square. Using the technique of completing the square, write an explanation of how to solve the following equation that you could give to the student who had been absent: x + 8x 3 = 0 Ans: x = 4 ± ) If one root of a quadratic equation is 1 and the equation is x + bx + 3 = 0, what is the other root? Ans: 3 7

8 Do Now: (Questions 1 & ) 1.) Which expression shows the simplified form of 34 i? (1) 1 (3) i () 1 (4) i Day Graphing and Operations with Complex Numbers.) Simplify: i + 13i Steps to Graph Complex Numbers: Complex Numbers are always of the form a + bi, where a and b are real numbers and i is the imaginary unit. 1.) The a is the x-value, and the b is the y-value. Graph the point (a, b)..) The i tells you that it is not only a point, but a vector. You must draw an arrow from the origin (0, 0) to the point that you graphed. 3.) Label the vector a + bi. Directions: Graph each complex number. 3.) Graph the complex number 3+ 4i. 4.) Graph the complex number 4+ 5i. 5.) Graph the complex number 6 7i. 6.) Graph the complex number 9 i. 8

9 Steps to Adding/Subtracting Complex Numbers: Complex Numbers are always of the form a + bi, where a and b are real numbers and i is the imaginary unit. 1.) Combine like terms. You can only combine imaginary numbers with other imaginary numbers..) Remember, if you are subtracting, you must distribute the negative. Directions: Perform the indicated operation and express in simplest a + bi form. 7.) (6 + 7i) + (1 + i) 8.) (3 5i) + ( + i) 9.) ( 3 + 3i) (1 + 5i) 10.) ( 8 i) ( + 8 i) Steps to Finding the Additive Inverse of a Complex Number: Change all signs, on a and on b. Ex) a + bi a bi Directions: Find the additive inverse of each expression. 11.) 6 + i 1.) 1+ 3i 13.) 1 9i 14.) 8i Steps to Finding the Conjugate of a Complex Number: Change the middle sign, on b only. Ex) a + bi a bi Directions: Find the conjugate of each expression. 15.) 6 + i 16.) 1+ 3i 17.) 1 9i 18.) 8i 9

10 Steps to Multiplying Complex Numbers: 1.) FOIL (or calculator!).) Remember, i = 1, so i = 1. Directions: Perform the indicated operation and express the answer in simplest a + bi form. 19.) (5 + i)(3 + 4 i) 0.) (3 5 i)( + i) 1.) (3 + i)(3 + 3 i).) ( + 6 i) 3.) (9 + 3 i)(9 3 i) 4.) (1 9)( i 1+ 9) i 10

11 Day Graphing and Operations with Complex Numbers HOMEWORK Directions (Questions 1 3): Graph each complex number. 1.) 6 + 7i.) 3i 3.) 3 4i Directions (Questions 4 9): Find each sum or difference of the complex numbers in a + bi form. 4.) (1+ 9 i) (1+ i) 5.) (10 1 i) (1 + 7 i) 6.) (5 6) i + (4+ ) i Ans: 0+ 7i 7.) (4+ 1) i + ( 4 ) i Ans: 19i 8.) ( + i) + ( i) 4 4 Ans: 9 4i 9.) 3 i+ (1+ 4 i) Ans: i Ans:.75.5i or i Directions (Questions 10 17): Find each product in simplest a + bi. 10.) (1+ 9 i)(1+ i) 11.) (10 1 i)(1 + 7 i) Ans: 1+ 7i Ans: i 11 Ans: 04 74i

12 1.) (3 + 5 i)( + i) 13.) 3 i(1+ 4 i) 14.) ( 4 4 i)( 4 + i) Ans: 1+ 13i 15.) ( 1 i)(1 i) Ans: 1 + 3i 16.) ( i)(1+ i) Ans: 0 + 1i 17.) 3 1 (3 i)( + i) Ans: i Ans: 0 5i Ans: 1 + 0i Directions (Questions 18-1): For each question, find a) the additive inverse and b) the conjugate. 18.) 3 + i 19.) 1 i 0.) i Ans: a) 3 i, b)3 i 1.) π + i Ans: a) 1+ i, b) 1+ i Ans: a) i, b) i 1 Ans: a) π i, b) π i

13 Review.) One of the roots is given. Find the other root and the value of c. x + 3x + c = 0; one root = 1 Ans: x=.5, c= 5 13

14 Do Now: Questions 1 & 1.) The expression ( ) 3 7i is equivalent to (1) i (3) 40 4i () i (4) 58 4i Day 3 Dividing Complex Numbers.) The product of ( 1 + i) and ( i) 1 is equal to? (1) 1 i (3) 3 i () 3 + i (4) 1+ 4i Steps to Finding the Multiplicative Inverse of a Complex Number: Complex Numbers are always of the form a + bi, where a and b are real numbers and i is the imaginary unit. 1.) The multiplicative inverse is found by flipping the fraction. (i.e., put a 1 in the numerator).) Similar to radicals, you can t have an imaginary number in the denominator, so you must rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator. 3.) Simplify the resulting fraction to simplest a + bi form. Directions: For each problem, find the multiplicative inverse in simplest a + bi form. 1.) 4i.) 3 + 4i 14

15 Steps to Dividing Complex Numbers: Monomial Denominator Multiply top and bottom by the identical term. Binomial Denominators 1.) Similar to radicals, you can t have an imaginary number in the denominator, so you must the denominator by multiplying both the numerator and denominator by the of the denominator..) FOIL top and bottom separately. (calculator i-parts) 3.) Simplify the resulting fraction to simplest a + bi form. Directions: For each problem, divide and express the answer in simplest a + bi form i 3.) 3i 4.) + 3i 1+ i Practice Problems Directions: For each problem, divide and express the answer in simplest a + bi form. 5 i 5.) 5+ i 6.) i 1+ i 15

16 Day 3 Dividing Complex Numbers HOMEWORK **Complete any Practice Problems from class work that have not been completed** Directions: For each problem, divide and express the answer in simplest a + bi form. i 1.) 3 + i.) + 9i 1 i Ans: 1 1 i 3.) 4 6i 5i Ans: i 4.) 1 5i Ans: i 16 Ans: i

17 5.) 3 i i Ans: 3 1 i 6.) 3 + i Review x 3 x ) For what value(s) of x does =? 3 Ans: i 8.) Write + 3 with a rational denominator. 3 Ans: Ans:

18 Do Now: (Questions 1-) 1.) What is the quotient when ( + 5i) is divided by ( i) 7 + 7i (1) (3) i () (4) 7 Day 4 Complex Roots of Quadratic Equations 1?.) Divide and simplify in simplest a + bi form: 10 5i + 6i Directions: Solve each quadratic equation by the method given. 3.) Find all roots of the equation x + 4x+ 13 = 0, using the method of completing the square. 18

19 4.) Find the zeros of the equation 4x 1x+ 5 = 0, using the quadratic formula. 5.) Solve for x in simplest radical form: x + 9= 8 x 19

20 Day 4 Complex Roots of Quadratic Equations HOMEWORK Directions: Use the quadratic formula to find the imaginary roots, in simplest radical form if possible. 1.) x 4x+ 8= 0.) x + 13 = 4x Ans: ± i 0 Ans: ± 3i

21 3.) Solve the equation x 6x= 1 and express the roots in simplest a + bi form. Ans: 3± i 3 Directions: Use the method of completing the square to find the imaginary roots, in simplest radical form. 4.) x + 6x+ 10 = 0 1 Ans: 3± i

22 5.) x = x 10 6.) x + 17 = 8x Ans: 1± 3i Ans: 4 ± i

23 Review 7.) Write in simplest radical form: 64x + 40x x 4 8.) Divide and simplify: 8 + 6i i Ans: x Ans: 3 4i 3

24 Do Now: QUIZ Day 5 Nature of the Roots (1) The discriminant is an expression that determines the nature of the roots of a quadratic equation, and the preferable method of solving the equation. Ways to Describe the Roots of a Quadratic Equation (1) Real, Rational, Unequal () Real, Irrational, Unequal (3) Real, Rational, Equal (4) Imaginary Case 1: DISCRIMINANT b 4ac Case 4: Case : Case 3: 4

25 Directions: Describe the nature of the roots. (1) Real, Rational, Unequal; () Real, Irrational, Unequal; (3) Real, Rational, Equal; (4) Imaginary.) x 4x+ 3= 0 3.) x 3x 5= 0 4.) x 5= 0 5.) x 4x+ 4= 0 6.) x x+ 5= 0 7.) x x= 4 5

26 Day 5 Nature of the Roots (1) HOMEWORK Directions: Find the value of the discriminant and determine if the roots of the quadratic equation are (1) Rational and unequal; () Rational and equal; (3) Irrational and unequal; or (4) Imaginary. 1.) x 1x+ 36 = 0.) x + 7x= 0 3.) x + 3x+ 1= 0 Ans: () 4.) x 8= 0 Ans: (1) 5.) 4x x= 1 Ans: (3) 6.) 3x = 5x 3 Ans: (1) Ans: (3) Ans: (4) 6

27 Review 7.) Solve the quadratic equation by completing the square and express your answer in simplest radical form, if possible. x + 6x+ 4= 0 8.) Solve the quadratic equation 3x 6x= 1 using the quadratic formula. a. Express your answer in simplest radical form. b. Write, to the nearest tenth, a rational approximation for the roots. Ans: x = 3± 5 7 Ans: a) 3 ± or 1±, b) {., 0.} 3

28 Do Now: (Questions 1 & ) 1.) x 4x+ 4= 0.) x + 6x+ 3= 0 Day 6 Nature of the Roots () a) Find the value of the discriminant. a) Find the value of the discriminant. b) Describe the nature of the roots: (1) rational and unequal () rational and equal (3) irrational and unequal (4) imaginary c) Describe the graph: (1) Crosses in two distinct rational places () Crosses in two distinct irrational places (3) Tangent to the x-axis (4) Lies entirely above the x-axis (5) Lies entirely below the x-axis b) Describe the nature of the roots: (1) rational and unequal () rational and equal (3) irrational and unequal (4) imaginary c) Describe the graph: (1) Crosses in two distinct rational places () Crosses in two distinct irrational places (3) Tangent to the x-axis (4) Lies entirely above the x-axis (5) Lies entirely below the x-axis Equations Steps to Find the Missing Value Given the Nature of the Roots: 1.) Find your a, b, and c values based on the standard form ax + bx + c = 0..) Plug into the discriminant. If you are told your roots are equal, set the discriminant equal to zero and solve for the missing value. If you are told your roots are real, set the discriminant greater than or equal to zero and solve for the missing value. If you are told your roots are imaginary, set the discriminant less than zero and solve for the missing value. Equal Real Imaginary b 4ac = 0 b 4ac 0 b 4ac < 0 Directions: Find the missing value. 3.) Find the value of k if the roots of the equation x 6x+ k = 0 are equal. 8

29 4.) Find the values of k which will make the roots of the equation real: x 8x+ k = 0 5.) Find the values of k for which the roots of the equation will be imaginary: kx 1x + 4 = 0 Practice Problems 6.) Find the value of k which will make the roots of the equation equal: x + 8x= k 7.) Find the value of k which will make the roots of the equation real: kb 9b = 4 8.) Find the values of k for which the roots of the equation will be imaginary: kc + 10c + 1 = 0 9.) Find the value of k which will make the roots of the equation equal: y + ky + 36 = 0 9

30 10.) Find the values of k which will make the roots of the equation real: x + 8x+ k = 0 11.) Find the value of k which will make the roots of the equation equal: kx + 4x + 1= 0 30

31 Directions: Find the missing value. 1.) For what value of k are the roots of x 8x+ k = 0 equal? Day 6 Nature of the Roots () HOMEWORK.) Find the values of k which will make the roots of the equation real: x 1x+ k = 0 Ans: k = 8 3.) Find the values of k for which the roots of the equation will be imaginary: y + k = 8y Ans: k 36 4.) Find the value of k which will make the roots of the equation equal: kx 10x + 5 = 0 Ans: k > 16 5.) Find the values of k which will make the roots of the equation real: ky + 8y + 4= 0 Ans: k = 5 6.) Find the values of k which will make the roots of the equation imaginary: x + 6x+ k = 0 Ans: k 4 31 Ans: k > 9

32 Review 7.) Write the expression in simplest form: xy x y 8.) Solve for x in simplest a + bi form: x 4x+ 4= 0 Ans: 3x y Ans: 1± i 3

33 1.) The roots of the quadratic equation 3x x 3= x 5x 7 are (1) real, rational, and equal () real, rational, and unequal (3) real, irrational, and unequal (4) imaginary Day 7 Review.) If a quadratic equation with real coefficients has a discriminant whose value is 5, then the two roots must be (1) real, rational, and equal () real, rational, and unequal (3) real, irrational, and unequal (4) imaginary 3.) Which might be the value of a discriminant of a quadratic equation whose graph lies entirely above the x-axis? 4.) In simplest form, what is the sum of 38 11i? 10 4i and (1) -10 () 0 (3) 6 (4) 9 5.) In simplest form, what is the product of 3 and 4i? 10 i 6.) Simplify:

34 7.) Graph the sum of 4 + 5i and 7i. 8.) Find the values of k for which the roots of the equation will be imaginary: x x+ k = 0 9.) Find the roots in simplest a + bi form: 3x 6x = Directions: Perform the indicated operation and express in simplest a + bi form. 10.) (3 + 5i) + (1 + i) 11.) ( + 5i) (1 + 5i) 34

35 13.) What is the multiplicative inverse of 8 3i? 1.) ( + 3i)( 4 6i) 14.) What is the additive inverse of 6 + 5i? 15.) What is the conjugate of 4 i? 16.) 5i 17.) If one root of a quadratic equation is 6 + i, 6 + i find the equation. 35