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WARM-UP ο‘ Factor the following expressions: ο‘ π π₯ = 3π₯ 2 β 8π₯ + 4 ο‘ π π₯ = 15π₯ 2 β 27π₯ β 6 4.4 COMPLEX NUMBERS https://www.khanacademy.org/math/precalcul us/imaginary-and-complex-numbers/theimaginary-numbers/v/introduction-to-i-andimaginary-numbers As you watch the video please take notes on things that you may believe to be important! NOTES FROM THE VIDEO THE SQUARE ROOT OF A NEGATIVE NUMBER. π= 2 π = 3 π = 4 π = The only values youOF need two! POWERS I to remember are the first You can always break down a high power into (π 2 )π β π β’ π5 = β’ π6 = β’ π7 = β’ π8 = ο‘Simplify. ο‘a) i 20 ο‘c) i 17 EXAMPLE 2 b) i 43 d) i 34 THE SQUARE ROOT OF A NEGATIVE NUMBER. Property Example 1. If r is a positive real number, then οr ο½ i r . ο5 ο½ i 5 2. By Property (1), it follows that ο¨i r ο© ο½ οr 2 ο¨i 5 ο© ο½ i 2 2 ο 5 ο½ ο5 EXAMPLE 1 ο‘Simplify. ο‘1) β180 WARM UP 2) i 85 HOMEWORK ο‘ Simplifying Imaginary Number worksheet HOMEWORK ANSWERS ο‘1.6i ο‘2.10i ο‘3.9i ο‘4.-14i ο‘13. π 3 ο‘14. π 29 ο‘15. 3π 11 ο‘16. -π 10 HOMEWORK ANSWERS ο‘ ο‘ ο‘ ο‘ ο‘ 5. i 6.-2i 19. 10i 3 20.10i 2 22.-5i 3 ο‘ 24.4i ο‘ ο‘ ο‘ ο‘ ο‘ ο‘ ο‘ 1 8 Back: 7.1 8.-1 12.i 13.not possible 14.-i 15.6i ο‘ https://www.khanacademy.org/math/precalculus/imaginary and-complex-numbers/the-complex-numbers/v/complexnumber-intro COMPLEX NUMBERS complex number: a + bi ο‘ a is the real part ο‘ bi is the imaginary part Real numbers: β1 , 3, π , π, 2/5 Complex Numbers: (a + bi, b β 0), 2 + 3i, 5 β 5i Pure Imaginary Numbers: (0 + bi, b β 0), β4i, 6i Add/subtract the real parts to get the real part, add/subtract the imaginary parts to get the imaginary part ο‘ a. (β1 + 2i) + (3 + 3i) ο‘ b. (2 β 3i) β (3 β 7i) Add/subtract the real parts to get the real part, add/subtract the imaginary parts to get the imaginary part ο‘ C.)(10-7i) + (6+9i) ο‘ D. (-3+i) + (-4-i) EXAMPLE 5 ο‘ Find the values of x and y that make the equation 3x + 4yi = 9 + 12i true. Multiply the imaginary numbers. ο‘ a. βi(3 + i) ο‘ c. (5 + i)(3 β 7i) Multiply the imaginary numbers. ο‘ D. (1+2i)(11-4i) ο‘ E. (4-i)(6-6i) EXAMPLE 5 ο‘ Find the values of x and y that make the equation 2x + yi = β14 β 3i true. Multiply the imaginary numbers. ο‘ d. (2 + 3i)(2 β 3i) Multiply the imaginary numbers. ο‘ d. (4 - 6i)(4 + 6i) ο‘a + b i and a β b i are conjugates. ο‘When you multiply conjugates you get a real number β no imaginary parts EXAMPLE 4 ο‘ Divide the complex numbers. Answer should be in standard form. ο‘ a. 2 ο 7i 1ο« i EXAMPLE 4 ο‘ Divide the complex numbers. Answer should be in standard form. ο‘ b. 3 ο« 11i ο1 ο 2i EXAMPLE 4 ο‘ Divide the complex numbers. Answer should be in standard form. ο‘ b. ο3 ο i 3ο«i HOMEWORK ο‘ Page 250: 26-31 and 43-47 HOMEWORK ANSWERS ο‘ ο‘ ο‘ ο‘ ο‘ ο‘ ο‘ ο‘ 26. 16+2i 27. -7 28. 3+7i 29. 9 30. 1 31 . 30+17i 43. x=-2, y=-3 46. 3x 2 = -48 2x 2 = -10 Solve 5y 2 + 20 = 0. 2x 2 + 26 = β10 4x 2 + 4 = 0 9x 2 + 4 = 0 EXAMPLE 9 2 3 ο¨ x ο 4 ο© ο« 7 ο½ 31 EXAMPLE 8 1 2 ο 2 ο¨x ο« 1ο© ο½5 WRITING QUADRATIC FUNCTIONS WITH COMPLEX ROOTS ο‘ imaginary solutions come in pairs ο‘ if an imaginary number is a zero, then itβs conjugate is also a zero ο§ so, if 4 + 3i is a solution, then 4 β 3i is also a solution Write a quadratic function that has real coef ficients and a leading coef ficient of 3 and has a complex root of 2 i. Write a quadratic function that has real coef ficients and a leading coef ficient of 1 and has a complex root of 4 + i. TICKET OUT Write a quadratic function that has real coef ficients and a leading coef ficient of 1 and has a complex root of 3- i.