Chapter 2 – Polynomial and Rational Functions

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Presentation transcript:

Chapter 2 – Polynomial and Rational Functions 2.4 – Complex Numbers

Some quadratic equations have no real solutions but still can have imaginary solutions. and Evaluate to fill in the tables. Look for a pattern: i0 i1 i2 i3 i4 i5 i6 1 i7 i8 i9 i10 i11 i12 i13 -i i 1 -1 i -1 -i -i 1 i 1 -1 i Imaginary Numbers

Complex #s = the set of all real and all imaginary numbers. If a and b are real numbers, then a complex number can be written in the standard form of a + bi. a is the real part bi is the imaginary part Ex: Simplify (3 – 2i) – (2 – 3i). CLT (combine like terms)! 3 – 2i – 2 + 3i 1 + i Complex Numbers

Multiplying Complex Numbers Ex: Simplify . Write in i form and multiply! (2i)(4i) = 8i2 = 8(-1) = -8 Ex: Simplify (3 + 4i)(2 – i). FOIL to get 6 – 3i + 8i – 4i2 = 6 + 5i – 4(-1) = 10 + 5i Ex: Simplify (5 + 2i)2 = 25 + 20i + 4i2 = 25 + 20i + 4(-1) = 21 + 20i Multiplying Complex Numbers

Simplify: (3 – 4i)(2 + 7i) 6 – 28i 6 – 15i 34 + 13i -22 + 13i 17 + 15i

Simplify: (9 – 2i)(9+2i) 77 83 81 – 4i 83 + 36i 85

Complex Conjugates The complex conjugate of a + bi is a – bi. Multiplying two complex conjugates results in a real number: (a + bi)(a – bi) = a2 + b2 Use this fact to get complex numbers out of denominators! Ex: Write in standard form: Multiply the entire fraction by the complex conjugate of the denominator! Complex Conjugates

Simplify: (3 – 3i)/(4+2i)

The Complex Plane When plotting complex coordinates, remember: The x-axis is the real axis The y-axis is the imaginary axis Plot the following points: 2 + 4i 1 – 2i 3 -i The Complex Plane