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**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

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**Warm-up Divide the following using Long Division:**

(6x3 - 16x2 + 17x - 6) (3x –2 ) 2x2 – 4x + 3

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**Warm-up Divide the following with Synthetic Division**

(5x3 – 6x2 + 8) (x – 4)

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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)

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**Complex Numbers Section 2-4**

Digital Lesson Complex Numbers Section 2-4

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

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Complex Numbers Real Numbers Imaginary Numbers Rational Irrational

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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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The Imaginary Unit

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Negative Radicals

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Negative Radicals

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**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 Add or Subtract Complex Numbers

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**Adding Complex Numbers**

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

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**Subtracting Complex Numbers**

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

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**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. Product of Complex Numbers

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

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Homework WS 3-7

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Chapter 5 Section 4: Complex Numbers. VOCABULARY Not all quadratics have real- number solutions. For instance, x 2 = -1 has no real-number solutions because.

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