2.4 complex numbers

In this chapter we solve complex equations by adding, subtracting, multiplying, and dividing.
we use the imaginary number i which is defined as the square root of -1.


If a and b are are real numbers, the number a + bi is a complex number. which is supposed to be written in standard form a+bi = c+di
iff a=c and b=d

If b is not = to zero, the number a+bi is an imaginary number

(4+4i) + (3-6i) = 7-2i addition of complex numbers

(4+4i) - (3-6i) = 1+10i subtraction of complex numbers

(4+4i)(3-6i) = 12 -12i -24i^2 i^2 =-1 -24(-1) =24 multiplication of complex numbers
= 36-12i

4+4i/3-6i multiply numerator and denominator by 3+6i to cancel out middle term
= -12+36i/9-36i^2

-12/45 + 36/45i
= -4/15+ 4/5 i (written in standard form) division of complex numbers


Expanding
zeros: 2, 3-4i, and 3+4i

f(x)= (x-2)(x-(3-4i))(x-(3+4i))
f(x)= (x-2)(x-3+4i)(x-3-4i)
(a + b) (a - b) = a^2 -b^2
f(x)= (x-2)(x-3)^2-(4i)^2

f(x)= x^3-8x^2+37x-50 (expanded)