Problem 61 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

The product is \(-8 - 6i\).

Step 1: Use the Distributive Property

Apply the distributive property (FOIL method) to the expression \((a+b)(c+d) = ac + ad + bc + bd\). For the given expression \((3+i)(-3-i)\), distribute each term:- First: \(3 \times -3 = -9\)- Outer: \(3 \times -i = -3i\)- Inner: \(i \times -3 = -3i\)- Last: \(i \times -i = -i^2\)

Step 2: Simplify the Expression

Combine like terms:- Combine \(-3i\) and \(-3i\) to get \(-6i\).- Recognize that \(-i^2\) is equal to \(-(-1)\) or \(1\) because \(i^2 = -1\).So, we have:\(-9 - 6i + 1\).

Step 3: Write in Standard Form

Combine the real parts:\(-9 + 1 = -8\).The standard form for the expression is \(-8 - 6i\).

Key Concepts

Distributive Property

The distributive property is a key algebraic rule that allows us to multiply a single term by two or more terms inside a parenthesis. It establishes how each element in one bracket multiplies with each element in the other.
For example, in \(a + b\)(c + d) = ac + ad + bc + bd\, each term is distributed across the others. This ensures each term in the first bracket multiplies with every term in the second. When applying this to complex numbers such as \(3+i\)(-3-i)\, this technique helps to systematically simplify the expression.

The distributive property's ability to handle complex numbers allows us to transform a product of sums into simpler terms, making calculations easier and more manageable.

FOIL Method

The FOIL method is a specific use of the distributive property commonly applied to multiply binomials. FOIL stands for First, Outer, Inner, Last, describing the order in which you multiply the terms:

• First: Multiplying the first terms in each binomial.
• Outer: Multiplying the outer terms.
• Inner: Multiplying the inner terms.
• Last: Multiplying the last terms.

When applied to \(3+i\)(-3-i)\, each pairing of terms is calculated:

• exttt{-9}: the product of the first terms (3 and -3).
• exttt{-3i} and exttt{-3i}: the products of the outer (3 and -i) and inner (i and -3) terms respectively.
• exttt{-i^2}: the product of the last terms (i and -i).

Using FOIL helps organize multiplication when complex numbers are involved, providing a straightforward approach to expand expressions.

Simplifying Expressions

Simplifying expressions means reducing them to their simplest form by combining like terms and applying mathematical rules.
In our expression \(-9 - 6i + 1\), simplifying involves:

• Combining the constants \-9\ and \+1\, which results in \-8\.
• Recognizing that \-i^2\ equals \+1\ because \i^2 = -1\.
• Adding the like terms \-3i\ and \-3i\ to obtain \-6i\.

Simplifying is crucial in reaching the cleanest possible form of an expression, making it easier to interpret or further manipulate.

Standard Form

The standard form for complex numbers is expressed as \a + bi\, where \a\ and \b\ are real numbers. Here, \a\ represents the real part and \bi\ the imaginary part.
To write \(-9 - 6i + 1\) in standard form, you first combine the real components \-9\ and \+1\ to form \-8\. This leads to a final expression of \-8 - 6i\.
Using standard form is important for clarity and consistency, especially when dealing with complex expressions. It ensures that the expression is easily understandable and can be used correctly in further calculations.