Problem 27 - Algebra 2 and Trigonometry

Question

In \(26-37,\) find each product. $$ (3+2 i)(3+3 i) $$

Step-by-Step Solution

Answer

The product is \(3 + 15i\).

Step 1: Apply the Distributive Property

To find the product \((3+2i)(3+3i)\), apply the distributive property (also known as FOIL for binomials). This means we will multiply each term in the first binomial by each term in the second binomial.So, we calculate:1. The product of the first terms: \(3 \times 3 = 9\)2. The product of the outer terms: \(3 \times 3i = 9i\)3. The product of the inner terms: \(2i \times 3 = 6i\)4. The product of the last terms: \(2i \times 3i = 6i^2\)

Step 2: Calculate \(i^2\) and Combine Like Terms

Recall that \(i^2 = -1\). Therefore, \(6i^2 = 6(-1) = -6\).Now, combine all the results obtained from Step 1:\(9 (from \: 3 \times 3) + 9i (from \: 3 \times 3i) + 6i (from \: 2i \times 3) + -6 (from \: 2i \times 3i)\).Combine the like terms and simplify:- Real part: \(9 - 6 = 3\)- Imaginary part: \(9i + 6i = 15i\)So the product is \(3 + 15i\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply a single term across each term in a binomial, or between two binomials as we often see with complex numbers. When you apply this property to multiply two binomials like

• (3 + 2i)(3 + 3i), you are expanding the expression by multiplying each part of one binomial by every part of the other binomial.

In practice, this refers to the FOIL method, especially useful with binomials:

• First: Multiply the first terms in each binomial: 3 × 3 gives 9.
• Outer: Multiply the outer terms: 3 × 3i gives 9i.
• Inner: Multiply the inner terms: 2i × 3 gives 6i.
• Last: Multiply the last terms: 2i × 3i gives 6i².

This method ensures that you don’t miss any term and is particularly powerful when dealing with expressions involving complex numbers. The results are then simplified by combining like terms, especially noting that any term related to \(i^2\) needs to be further simplified.

Imaginary Unit

When working with complex numbers, one crucial component is the imaginary unit, which is denoted as \(i\). This unit is defined by the property \(i^2 = -1\).

This characteristic is fundamental when simplifying complex number expressions, especially in multiplication. For example, in the expression

• 2i × 3i = 6i², the \(i^2\) must be converted according to its basic definition.

This translates the product into a real number:

• 6i² = 6(-1) = -6.

Understanding \(i\) and \(i^2\) is essential in handling complex numbers because it allows you to seamlessly transform imaginary products into real numbers, simplifying the entire process of calculation.

Binomials

A binomial is a simple algebraic expression involving two terms that are usually joined by addition or subtraction. It often appears as a + b or a - b.

In the context of complex numbers, binomials incorporate both a real and an imaginary part, such as:

• (3 + 2i) and (3 + 3i).

When you multiply binomials, particularly those including imaginary components, you resort to the distributive property to effectively manage each pair of terms.
This is where the FOIL method—emphasizing First, Outer, Inner, Last—shines, ensuring comprehensive term-by-term multiplication.
Once every multiplication is complete, the expression reduces by combining like terms, transforming the equation into a single complex number.
The outcome gives both the real part and the imaginary part, demonstrated in the exercise solution: 3 + 15i. By understanding binomial products, you can confidently tackle complex equations involving these vital mathematical building blocks.