LESSON 13 POLYNOMIAL EQUATIONS COMPLEX NUMBERS Pre Calculus

LESSON 13 - POLYNOMIAL EQUATIONS & COMPLEX NUMBERS Pre. Calculus - Santowski

LESSON OBJECTIVES 2/1/2022 Review the basic idea behind complex number system Review basic operations with complex numbers Find and classify all real and complex roots of a polynomial equation Write equations given information about the roots Pre. Calculus - Santowski 2

FAST FIVE http: //mathforum. org/johnandbetty/frame. htm Pre. Calculus - Santowski STORY TIME. . . 2/1/2022 3

(A) INTRODUCTION TO COMPLEX NUMBERS 2/1/2022 Solve the equation x 2 – 1 = 0 Pre. Calculus - Santowski We can solve this many ways (factoring, quadratic formula, completing the square & graphically) In all methods, we come up with the solution of x = + 1, meaning that the graph of the quadratic (the parabola) has 2 roots at x = + 1. Now solve the equation x 2 + 1= 0 4

(A) INTRODUCTION TO COMPLEX NUMBERS Long ago mathematicians decided that this was too restrictive They did not like the idea of an equation having no solutions -- so they invented them. They proved to be very useful, even in practical subjects like engineering. Pre. Calculus - Santowski the equation x 2 + 1= 0 has no roots because you cannot take the square root of a negative number. 2/1/2022 5

(A) INTRODUCTION TO COMPLEX NUMBERS So then, if or 6 Pre. Calculus - Santowski 2/1/2022 since solutions are “invented” then let’s “invent” a symbol to help us We shall use the symbol i to help us out and define i as …

(A) INTRODUCTION TO COMPLEX NUMBERS 2/1/2022 Note the difference (in terms of the expected solutions) between the following 2 questions: Pre. Calculus - Santowski Solve x 2 + 5 = 0 where 7

(B) OPERATIONS WITH COMPLEX NUMBERS 2/1/2022 Pre. Calculus - Santowski So if we are going to “invent” a number system to help us with our equation solving, what are some of the properties of these “complex” numbers? How do we operate (add, sub, multiply, divide) How do we graphically “visualize” them? Powers of i Absolute value of complex numbers 8

(B) OPERATIONS WITH COMPLEX NUMBERS 2/1/2022 Exercise 1. Add or subtract the following complex numbers using your TI-84 (switch to complex mode). Pre. Calculus - Santowski (a) (3 + 2 i) + (3 + i) (b) (4 − 2 i) − (3 − 2 i) (c) (− 1 + 3 i) + (2 + 2 i) (d) (2 − 5 i) − (8 − 2 i) Generalize HOW do you add complex numbers? 9

(B) OPERATIONS WITH COMPLEX NUMBERS z 1 + z 2 = a + ib + c + id = a + c + i(b + d) z 1 − z 2 = a + ib − c − id = a − c + i(b − d) Pre. Calculus - Santowski add or subtract two complex numbers, z 1 = a + ib and z 2 = c+id, the rule is to add the real and imaginary parts separately: 2/1/2022 to 10

(B) OPERATIONS WITH COMPLEX NUMBERS 2. Multiply the following complex numbers using your TI-84 (switch to complex mode). (3 + 2 i)(3 + i) (b) (4 − 2 i)(3 − 2 i) (c) (− 1 + 3 i)(2 + 2 i) (d) (2 − 5 i)(8 − 3 i) (e) (2 − i)(3 + 4 i) Generalize Pre. Calculus - Santowski (a) 2/1/2022 Exercise HOW do you multiply complex numbers? 11

(B) OPERATIONS WITH COMPLEX NUMBERS We multiply two complex numbers just as we would multiply expressions of the form (x + y) together Example (2 + 3 i)(3 + 2 i) = 2 × 3 + 2 × 2 i + 3 i × 3 + 3 i × 2 i = 6 + 4 i + 9 i − 6 = 13 i Pre. Calculus - Santowski (a + ib)(c + id) = ac + a(id) + (ib)c + (ib)(id) = ac + iad + ibc − bd = ac − bd + i(ad + bc) 2/1/2022 12

(C) COMPLEX CONJUGATION Pre. Calculus - Santowski any complex number, z = a+ib, we define the complex conjugate to be: . It is very useful since the following are real: = a + ib + (a − ib) = 2 a = (a + ib)(a − ib) = a 2 + iab − (ib)2 = a 2 + b 2 2/1/2022 For Exercise 4. Combine the following complex numbers and their conjugates and find and (a) If z = (3 + 2 i) (b) If z = (3 − 2 i) (c) If z = (− 1 + 3 i) (d) If z = (4 − 3 i) 13

(D) SOLVING EQUATIONS USING COMPLEX NUMBERS Solve x 2 + 2 x + 5 = 0 where Pre. Calculus - Santowski Note the difference (in terms of the expected solutions) between the following 2 questions: 2/1/2022 14

(D) SOLVING EQUATIONS USING COMPLEX NUMBERS x 2 – 2 x = -10 3 x 2 + 3 = 2 x 5 x = 3 x 2 + 8 x 2 – 4 x + 29 = 0 Now verify your solutions algebraically!!! Pre. Calculus - Santowski the following quadratic equations where Simplify all solutions as much as possible Rewrite the quadratic in factored form 2/1/2022 Solve 15

(D) SOLVING EQUATIONS USING COMPLEX NUMBERS (a) What is the other root? (b) What are the factors of the quadratic? (c) If the y-intercept of the quadratic is 6, determine the equation in factored form and in standard form. Pre. Calculus - Santowski One root of a quadratic equation is 3 i 2/1/2022 16

(D) SOLVING EQUATIONS USING COMPLEX NUMBERS (a) What is the other root? (b) What are the factors of the quadratic? (c) If the y-intercept of the quadratic is 6, determine the equation in factored form and in standard form. Pre. Calculus - Santowski One root of a quadratic equation is 2 + 3 i 2/1/2022 17

(D) SOLVING POLYNOMIAL EQUATIONS USING COMPLEX NUMBERS x 3 – 2 x 2 + 9 x = 18 x 3 + x 2 = – 4 x 4 x + x 3 = 2 + 3 x 2 Now verify your solutions algebraically!!! Pre. Calculus - Santowski the following polynomial equations where Simplify all solutions as much as possible Rewrite the polynomial in factored form 2/1/2022 Solve 18

(D) SOLVING POLYNOMIAL EQUATIONS USING COMPLEX NUMBERS Pre. Calculus - Santowski the following polynomial functions State the other complex root Rewrite the polynomial in factored form Expand write in standard form 2/1/2022 For (i) one root is -2 i as well as -3 (ii) one root is 1 – 2 i as well as -1 (iii) one root is –i, another is 1 -i 19