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Warm-up

Warm-up. Divide the following using Long Division: (6x 3 - 16x 2 + 17x - 6)  (3x –2 ) Divide the following with Synthetic Division (5x 3 – 6x 2 + 8) (x – 4) Given the following polynomial and one of its factors, Find the remaining factors

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Warm-up

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  1. Warm-up • Divide the following using Long Division: • (6x3 - 16x2 + 17x - 6)  (3x –2 ) • Divide the following with Synthetic Division • (5x3 – 6x2 + 8) (x – 4) • Given the following polynomial and one of its factors, Find the remaining factors • (3x3 + 2x2 –19x + 6) : (x + 3) is a factor

  2. Warm-up • Divide the following using Long Division: • (6x3 - 16x2 + 17x - 6)  (3x –2 ) • 2x2 – 4x + 3

  3. Warm-up • Divide the following with Synthetic Division • (5x3 – 6x2 + 8) (x – 4)

  4. Warm-up • Given the following polynomial and one of its factors, Find the remaining factors • (3x3 + 2x2 –19x + 6) : (x + 3) is a factor • (x – 2)(3x – 1)

  5. Digital Lesson Complex NumbersSection 2-4

  6. Objectives • I can use “i” to write complex numbers • I can add, subtract, and multiply complex numbers • I can simplify Negative Square Roots

  7. Applications • Impedance readings for electronics and electrical circuits are all measured in complex units

  8. Complex Numbers Real Numbers Imaginary Numbers RationalIrrational

  9. Complex Numbers The set of all numbers that can be written in the format: a + bi ; “a” is the real number part “bi’ is the imaginary part

  10. The Imaginary Unit

  11. Negative Radicals

  12. Negative Radicals

  13. Add or Subtract Complex Numbers To add or subtract complex numbers: 1. Write each complex number in the form a + bi. 2. Add or subtract the real parts of the complex numbers. 3. Add or subtract the imaginary parts of the complex numbers. (a + bi) + (c + di) = (a + c) + (b + d)i (a + bi) – (c + di) = (a – c) + (b – d)i

  14. Adding Complex Numbers Example: Add (11 + 5i) + (8 – 2i ) = (11 + 8) + (5i – 2i ) Group real and imaginary terms. = 19 + 3i a + bi form

  15. Subtracting Complex Numbers Examples: Subtract: (– 21 + 3i ) – (7 – 9i) = (– 21 – 7) + [(3 – (– 9)]i Group real and imaginary terms. = (– 21 – 7) + (3i + 9i) = –28 + 12i a + bi form

  16. Product of Complex Numbers The product of two complex numbers is defined as: (a + bi)(c + di ) = (ac – bd ) + (ad + bc)i 1. Use the FOIL method to find the product. 2. Replace i2 by – 1. 3. Write the answer in the form a + bi.

  17. Examples 1. 7i(11– 5i) = 77i– 35i2 = 77i– 35 (–1) = 35 + 77i 2. (2 + 3i)(6 – 7i) = 12 –14i+18i–21i2 = 12 + 4i–21i2 = 12 + 4i–21(–1) = 12 + 4i + 21 = 33 + 4i

  18. Homework • WS 3-7

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