Problem 61 - A Graphical Approach to Precalculus with Limits

Question

Multiply as indicated. Write each product in standand form. $$(3+i)(-3-i)$$

Step-by-Step Solution

Answer

\((-8 - 6i)\)

Step 1: Distribute the terms

To start, apply the distributive property by multiplying each term in the first binomial \((3 + i)\) by each term in the second binomial \((-3 - i)\). This results in four separate products: \(3 \cdot (-3)\), \(3 \cdot (-i)\), \(i \cdot (-3)\), and \(i \cdot (-i)\).

Step 2: Calculate each product

1. Multiply \(3\) and \(-3\) to get \(-9\). 2. Multiply \(3\) and \(-i\) to get \(-3i\). 3. Multiply \(i\) and \(-3\) to get \(-3i\).4. Multiply \(i\) and \(-i\) to get \(-i^2\). Since \(i^2 = -1\), this becomes \(-(-1) = 1\).

Step 3: Combine like terms

Add all the products together: \(-9 - 3i - 3i + 1\). Combine the like terms, specifically the imaginary parts \(-3i\) and \(-3i\):\[-9 + 1 - 3i - 3i = -8 - 6i\].

Step 4: Write in standard form

The standard form for a complex number is \(a + bi\). Thus, the result \(-8 - 6i\) is already in standard form, where \(a = -8\) and \(b = -6\).

Key Concepts

Binomials

A binomial is a type of algebraic expression involving two terms. These terms are generally separated by either a plus or a minus sign. In the context of complex numbers, binomials often take the form of

• \((a + bi)\)
• \((c + di)\)

Here, \(a\) and \(c\) are real numbers, while \(b\) and \(d\) are coefficients of the imaginary unit \(i\). The expression \((3 + i)\) is a perfect example of a binomial in the complex domain.
When multiplying binomials, each term in the first binomial must be multiplied by each term in the second binomial. This is a practical application of the distributive property in algebra.

Distributive Property

The distributive property is a guiding principle in algebra that allows us to simplify expressions. It is expressed as:

• \(a(b + c) = ab + ac\)

When applied to the multiplication of binomials, it ensures that each part of one expression interacts with each part of the other. For the expression \((3+i)(-3-i)\), we distribute terms as follows:

• \(3 \cdot (-3)\)
• \(3 \cdot (-i)\)
• \(i \cdot (-3)\)
• \(i \cdot (-i)\)

This property not only helps to simplify calculations but also brings clarity when working with complex numbers. The distribution step ensures that no interaction between terms is overlooked.

Imaginary Unit

The imaginary unit, represented by \(i\), is used in complex numbers and is defined by the equation \(i^2 = -1\). This definition is fundamental to understanding complex multiplication. In our example, one of the products we encounter is \(i \cdot (-i)\).
Initially, this looks like a regular multiplication of similar terms like \(x \cdot x \), which is \(x^2\). However, due to the special property of \(i\), we have:

• \(i \cdot (-i) = -i^2\)
• Given \(i^2 = -1\), then \(-i^2 = -(-1) = 1\)

This transformation from \(-i^2\) to \(1\) is crucial in our calculation, allowing us to combine terms and simplify the final expression \((-8 - 6i)\). The imaginary unit turns what might be a perplexing calculation into a straightforward process.