Answer
The multiplication of complex number requires the use of the FOIL method.
Work Step by Step
Consider the two complex numbers, $\left( a+bi \right)$ and $\left( c+di \right)$
To multiply the two complex numbers, use the FOIL method.
FOIL method to multiply the four terms of the product:
\[\left( a+b \right)\left( c+d \right)=\overbrace{ac}^{\text{F}}+\overbrace{ad}^{\text{O}}+\overbrace{bc}^{\text{I}}+\overbrace{bd}^{\text{L}}\]
F is the first terms of each binomial.
O is the outside terms or first term of the first binomial and second term of the second binomial.
I is the inside terms or second term of the first binomial and first term of the second binomial.
L is the last terms of each binomial.
The standard form of a complex number is $a+bi$, where $a$ is the real part and $b$ is the imaginary part.
Procedure to multiply the two complex numbers by FOIL method.
1. Multiply the first terms of each binomial.
2. Multiply the outside terms or multiply the first term of the first binomial with second term of the second binomial.
3. Multiply the inside terms or multiply second term of the first binomial with first term of the second binomial.
4. Multiply the last terms of each binomial.
5. Combine the real and imaginary part after multiplication.
6. Simplify the real parts and imaginary parts.
7. Express the answer in standard form.
For example,
Consider the complex numbers, $\left( 3+i \right)$ and $\left( -5+4i \right)$
Use the FOIL method.
$\begin{align}
& \left( 3+i \right)\left( -5+4i \right)=3\left( -5 \right)+3\cdot 4i+i\left( -5 \right)+i\cdot 4i \\
& =-15+12i-5i+4{{i}^{2}}
\end{align}$
The imaginary unit is $i=\sqrt{-1}$, where ${{i}^{2}}=-1$.
Replace the value ${{i}^{2}}=-1$.
$\left( -5+4i \right)\left( 3+i \right)=-15+12i-5i+4\left( -1 \right)$
Make a group of real and imaginary terms.
$\left( -5+4i \right)\left( 3+i \right)=-15-4+12i-5i$
Simplify the real and imaginary terms.
$\begin{align}
& \left( -5+4i \right)\left( 3+i \right)=\left( -15-4 \right)+\left( 12-5 \right)i \\
& =-19+7i
\end{align}$
Write the answer in standard form.
$\left( -5+4i \right)\left( 3+i \right)=-19+7i$
Therefore, the multiplication of two complex numbers follows the FOIL method.
