Problem 25 - Algebra and Trigonometry

Question

Evaluate the expression and write the result in the form a bi. $$ (7-i)(4+2 i) $$

Step-by-Step Solution

Answer

The result is \(30 + 10i\).

Step 1: Distribute the First Term

We need to distribute each term in the first complex number \((7 - i)\) across each term in the second complex number \((4 + 2i)\). Start by distributing 7:\[7 imes 4 + 7 imes 2i = 28 + 14i\]

Step 2: Distribute the Second Term

Now distribute \(-i\) across each term in the second complex number \((4 + 2i)\):\[-i \times 4 + (-i) \times 2i = -4i - 2i^2\]

Step 3: Simplify Using \(i^2 = -1\)

The term \(-2i^2\) can be simplified since \(i^2 = -1\). Thus:\[-2i^2 = -2(-1) = 2\]

Step 4: Combine Like Terms

Now combine all the real terms and imaginary terms:Real terms: \(28 + 2 = 30\)Imaginary terms: \(14i - 4i = 10i\)Thus, the expression simplifies to:\[30 + 10i\]

Key Concepts

Multiplying Complex Numbers

When you multiply complex numbers, you're essentially expanding the product of two binomials, but with an extra twist. Complex numbers have both a real and an imaginary part. For example, in the expression \((7 - i)(4 + 2i)\), we treat \(7\) and \(4\) as real parts and \(-i\) and \(2i\) as imaginary parts. To multiply these, follow a similar process to multiplying two binomials:

• Multiply each part of the first complex number by each part of the second.
• Use rules of imaginary numbers such as \(i^2 = -1\).

This approach will require taking each term in the first complex number and distributing it across each term in the second complex number, setting you up for the next steps.

Distributive Property

The distributive property is a foundational concept used in algebra to multiply terms efficiently. It tells us that for any three numbers \(a\), \(b\), and \(c\),
\(a(b + c) = ab + ac\). In the context of complex numbers, we apply this property to multiply each term of a binomial expression by each term of another binomial expression. For example, in the expression
\((7 - i)(4 + 2i)\), we first distribute \(7\) to both \(4\) and \(2i\), and then \(-i\) to both \(4\) and \(2i\). Following these steps results in

• \(7 \times 4 = 28\)
• \(7 \times 2i = 14i\)
• \(-i \times 4 = -4i\)
• \(-i \times 2i = -2i^2\)

This practice makes handling complex numbers more manageable.

Simplifying Complex Expressions

Simplifying complex expressions involves reducing them to a standard form, usually expressed as \(a + bi\). For complex numbers, this means you need to handle the multiplication of imaginary units properly. Remember, \(i\) represents \(\sqrt{-1}\), and the square of \(i\) is -1.In the expression
i.e., \(-2i^2\), simplify further using \(i^2 = -1\). This makes \(-2i^2 = 2\) because you are multiplying \(-2\) by \(-1\).Simplifying expressions with imaginary terms ensures consistency and helps to avoid mistakes, particularly when combining terms later. Always convert \(i^2\) to \(-1\) before proceeding with further simplification steps.

Combining Like Terms

Combining like terms is the last step when dealing with complex number expressions, and this ensures that you end with a neat expression \(a + bi\). After distributing and simplifying, you will have multiple real and imaginary terms that need to be combined.In our example:

• Real parts combine as: \(28 + 2 = 30\)
• Imaginary parts combine as: \(14i - 4i = 10i\)

This means the expression simplifies to \(30 + 10i\). Always group real numbers together and imaginary terms (those with \(i\)) together to find the simplest form of the expression.