imaginary numbers

complex division examples - 5+2i/3-4i

5+2i/3-4i would be multiplied by 3+4i/3+4i, FOIL, then add like terms and simplify any i^2 to -1

factoring sum of squares example - 25x^2 + 144

(5x)^2 + (12)^2 ; answer is (5x + 12i)(5x - 12i)

simplifying example - simplify the square root of -49

(square root of -1)(square root of 49) for an answer of 7i

when multiplying...

FOIL, change i^2 to -1 (BUT if number like -15i^2, simplifying would be -15(-1) , so keep the sign)

i^2

Square root of -1^2 (-1)

Solve by factoring

^ same steps, just with 0 product property

a+bi

a = x axis ; real portion bi= y axis ; imaginary portion

when subtracting...

distribute - sign to the right and simplify

example - i^30

divide 30 by 4, use remainder to determine the power (30 goes into 4 seven times, with a remainder of 2, so the answer is i^2 aka -1)

when factoring sum of squares ...

final answer must not be able to be more simplified

complex divisions

multiply by complex conjugate of denominator to eliminate imaginary

when adding...

regular adding

i

square root of -1 (i)

i^3

square root of -1^3 (-i)

i^4

square root of -1^4 (1)

simplify example 2 - simplify square root of negative 8

square root of 8 = 2 square root of 2, so the answer is 2i square root of 2 (the i comes from the -)