Question 131945
A conjugate of a complex number {{{a+bi}}} is formed by changing the sign on the imaginary part.  So the conjugate of {{{a+bi}}} is {{{a-bi}}}.


To simplify an expression with a complex denominator, you need to multiply the entire expression by 1 in the form of the conjugate of the denominator divided by itself.  That means if you have {{{(a+bi)/(c+di)}}}, you want to multiply by 1 in the form of {{{(c-di)/(c-di)}}}.


Your problem:


{{{(3-i) / (1+i)}}}


The conjugate of the denominator is {{{(1-i)}}}, so you need to multiply the entire expression by {{{(1-i)/(1-i)}}}, thus:


{{{((3-i)/(1+i))((1-i)/(1-i))}}}


Now multiply numerator times numerator and denominator times denominator just like multiplying any other pair of fractions.


Using FOIL, the  numerator product becomes: {{{3+3i-i+1=4+2i}}} (remember {{{i^2=-1}}} so {{{-i*i=-(i^2)=-(-1)=1}}})


The denominator product is easier because, by using the conjugate, you have the factors of the difference of two squares, so {{{(1+i)(1-i)=1-i^2=1-(-1)=2}}}


Putting the pieces back together we have {{{(4+2i)/2=2+i}}}


Done.