Question

Evaluate the product, and write the result in the form $a+bi$.

$\left(7-i\right)\left(4+2i\right)$

Short answer

The product of complex numbers is $30+10i$.

Step-by-step solution

Step 1. Concept of complex number.

A complex number is of the form $a+bi$. Where a and b are some real numbers and i is the imaginary number.

The number a is known as the real part of the complex number and b is known as imaginary part of the complex number.

Step 2. Concept of multiplication of complex numbers.

To multiply two complex numbers, multiply them as binomial expressions and use the result of imaginary number, $i^2=-1$.

The FOIL method of multiplying binomials is

$\left(a+b\right)+\left(c+d\right)=a\cdot c+a\cdot d+b\cdot c+b\cdot d$

The imaginary number i is defined as the square root of negative one. That is, $i=\sqrt{-1}$.

Step 3. Find the product.

To find the product of the given complex numbers, multiply them as binomials using FOIL method and simplify further by multiplying the numbers as shown below

$$\begin{gathered} \left(7-i\right)\left(4+2i\right)=7\cdot 4+7\cdot \left(2i\right)-i\cdot 4-i\cdot \left(2i\right) \\ =28+14i-4i-2i^2 \\ =28+10i-2\left(-1\right)\left(i^2=-1\right) \\ =28+10i+2 \\ =30+10i \end{gathered}$$

Thus, the product of the complex numbers is $30+10i$.