If I were to try and rationalize a complex number like $Z + 2W + 3 \over Z + 1$, when I multiply by the conjugate of Z to make the denominator a real number, do I have to write z(conjugate)/z(conjugate) or z(conjugate) + 1/z(conjugate) + 1 or z(conjugate) - 1/z(conjugate) - 1??
(I'm sorry idk how to format the conjugate can someone help edit)
If $Z,W\in\mathbb{C}$ (and $Z\neq-1$) and if you wish to express $\dfrac{Z+2W+3}{Z+1}$ has a quotient whose denominator is a real number, the natural option consists in multiplying both numerator and denominator with $\overline{Z+1}$, thereby getting$$\frac{Z+2W+3}{Z+1}=\frac{(Z+2W+3)\left(\overline{Z+1}\right)}{(Z+1)\left(\overline{Z+1}\right)}=\frac{(Z+2W+3)\left(\overline Z+1\right)}{\lvert Z+1\rvert^2}.$$