8.5 Complex Numbers 1 Simplify the following using the imaginary - PowerPoint PPT Presentation

8.5 Complex Numbers 1

Simplify the following using the imaginary number i when necessary .    4 50 25 a. b. c. Back to Table of Contents 8.5-1

For each complex number, name the real part and the imaginary part.   7 2 i 8 i a. b. Back to Table of Contents 8.5-2

Add or subtract the following complex numbers.       (7 4 ) i (6 5 ) i (2 8 ) i (6 3 ) i a. b. Back to Table of Contents 8.5-3

Multiply the following complex numbers.    5(2 6 ) i (4 7 )(2 i 9 ) i a. c.  3 (6 i 4 ) i b. Back to Table of Contents 8.5-4

   3 5 i 4 6 i 2 i Back to Table of Contents 8.5-6

Divide the following. Put all answers in the standard form of a complex number (a + bi).    9 21 i 2 5 i 8 5 i a. b. c.  3 3 2 i 3 i Back to Table of Contents 8.5-7

Solve the following equations. Give answers in the standard form of a complex number. m      2 2 16 x 3 x 20 a. b. Back to Table of Contents 8.5-8

Solve the following equations. Give answers in the standard form of a complex number.    3 2 t 6 t 10 t 0 c. Back to Table of Contents 8.5-8

7.1 Rational Functions

Find the domain of the following rational functions. Determine if the excluded values represent where a vertical asymptote appears in the graph.  7 x  f x ( ) a.  x 12 Back to Table of Contents 7.1-6

Find the domain of the following rational functions. Determine if the excluded values represent where a vertical asymptote appears in the graph.  x 8  ( ) g x b.   2 x 6 x 16 Back to Table of Contents 7.1-6

7.2 Simplifying Rational Expressions To simplify rational expressions 1. Factor the numerator and denominator 2. Divide out any common factors 3. Leave in factored form Simplify:   3 40 x ( 5)( 7) a a a. b.   12 x ( a 2)( a 5) 13

Simplify the following rational expressions.     2 ( a 2) 8 h 6 h 16 a. b.     2 ( a 2)( a 5) 4 4 h h   2 12 x 19 x 21   3 2 c. 15 x 25 x 140 x Back to Table of Contents 7.2-2

Simplify the following rational expressions. Special case – Sometimes we need to factor out a negative 1.  3( x 4) a.  5(4 x ) Back to Table of Contents 7.2-3