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1 Warm-up Divide the following using Long Division: (6x x x - 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 (3x 3 + 2x 2 –19x + 6) : (x + 3) is a factor

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2 Warm-up Divide the following using Long Division: (6x x x - 6) (3x –2 ) 2x 2 – 4x + 3

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3 Warm-up Divide the following with Synthetic Division (5x 3 – 6x 2 + 8) (x – 4)

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

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Complex Numbers Section 2-4 Digital Lesson

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

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

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8 Complex Numbers Real NumbersImaginary Numbers RationalIrrational

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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

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10 The Imaginary Unit

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11 Negative Radicals

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12 Negative Radicals

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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

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

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

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

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

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19 Homework WS 3-7

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