Question

How do you foil imaginary numbers?

Answer 1

Hint: To solve this question we are going to use the definition of foil. Along with it we will consider any two imaginary numbers either same or different and foil them just like we multiply two numbers in multiplication. We will also use the term ${{i}^{2}}=-1$ in this question so that we get a lead towards the right answer.

Complete step by step solution:
A foil is a method in which we do partition of any two different or same binomials. For example, we take (x + 4) (x + 5). Now, to foil these two binomials we will multiply them in such a way that they give the same result which it will give after the multiplication of two algebras. For this we will apply
$\begin{align}
  & \left( a+b \right)\left( a+b \right)=a\cdot a+b\cdot a+a\cdot b+b\cdot b \\
 & \Rightarrow \left( a+b \right)\left( a+b \right)={{a}^{2}}+2ab+{{b}^{2}} \\
\end{align}$
Therefore, (x + 4) (x + 5) results into $x\cdot x+4\cdot x+5\cdot x+4\cdot 5={{x}^{2}}+9x+20$. So, this resulting equation is called the process of foil.
Now, we will consider two imaginary numbers and foil them as above. Let us say that the general form of an imaginary number be a + bi and multiply it with a + bi. Thus, its foil will be as the following.
$\begin{align}
  & \left( a+bi \right)\left( a+bi \right)=a\cdot a+a\cdot bi+a\cdot bi+bi\cdot bi \\
 & \Rightarrow \left( a+bi \right)\left( a+bi \right)={{a}^{2}}+2abi+{{b}^{2}}{{i}^{2}} \\
 & \Rightarrow \left( a+bi \right)\left( a+bi \right)={{a}^{2}}+2abi-{{b}^{2}}\left[ \because {{i}^{2}}=-1 \right] \\
\end{align}$
So, this is how we foil two different or same imaginary numbers.

Note: We can also take two imaginary numbers which are (a + bi) (c + di). This can be done as follows,
$\begin{align}
  & \left( a+bi \right)\left( c+di \right)=a\cdot c+a\cdot di+bi\cdot c+bi\cdot di \\
 & \Rightarrow \left( a+bi \right)\left( c+di \right)=ac+\left( ad+bc \right)i-bd \\
\end{align}$
One should remember such words which can be used in mathematics for calculation. As here the question asked about foil instead of distribution. Therefore, to solve this question we should know about its definition. The best way to solve these types of questions is to multiply any two types of imaginary numbers. The foil is similar to multiplication but the steps in foil have names like outer and inner and first last. The first and last terms are the same but the inner and outer terms are basically represented as ba and ab respectively in $\left( a+b \right)\left( a+b \right)=a\cdot a+b\cdot a+a\cdot b+b\cdot b$.