Problem 60 - Elementary and Intermediate Algebra

Question

Multiply. Write all answers in the form \(a+b i\) See Example 6 $$ (3-i)(2+3 i) $$

Step-by-Step Solution

Answer

The final answer in the form \(a+bi\) is \(9+7i\)."

Step 1: Identify the Expressions

Multiply. Write all answers in the form \(a+b i\) See Example 6
$$
(3-i)(2+3 i)
$$

Step 2: Multiply

Apply the distributive property or FOIL method as appropriate.

Step 3: Result

The final answer in the form \(a+bi\) is \(9+7i\)."

Key Concepts

Distributive Property

In the world of mathematics, the distributive property is a fundamental concept that helps us simplify expressions and solve equations. When it comes to multiplying complex numbers, the distributive property is incredibly useful. It states that multiplying a single term by each term within a parenthesis, and then adding the results, gives an equivalent expression. This is exactly how we handle the multiplication of complex numbers, like in the exercise
o( (3-i)(2+3i)
Distribute each term in the first binomial to every term in the second binomial:

• Multiply the numbers in the first position: \(3 \times 2 = 6\)
• Multiply the numbers in the outer position: \(3 \times 3i = 9i\)
• Multiply the numbers in the inner position: \(-i \times 2 = -2i\)
• Multiply the numbers in the last position: \(-i \times 3i = -3i^2\)

Every term from the first binomial is "distributed" to each term of the second binomial. Once distributed, we combine like terms for the simplified solution.

FOIL Method

The FOIL method is a handy mnemonic to remember how to apply the distributive property specifically for binomials (expressions of the form \( (a+b)(c+d)\)). The acronym stands for First, Outer, Inner, Last, representing the order in which pairs of terms are multiplied:

• First: Multiply the first terms in each binomial.
• Outer: Multiply the outermost terms in the binomial multiplication.
• Inner: Multiply the innermost terms.
• Last: Multiply the last terms in each binomial.

Applying FOIL to the exercise
(3-i)(2+3i)
helps break down the process into manageable steps:

• First: \(3 \times 2 = 6\)
• Outer: \(3 \times 3i = 9i\)
• Inner: \(-i \times 2 = -2i\)
• Last: \(-i \times 3i = -3i^2\)

Each of these products is an essential part of the final solution. The FOIL method is especially useful because it organizes the operation sequence clearly, ensuring you don't miss any products.

Imaginary Numbers

When dealing with complex numbers, imaginary numbers come into play. An imaginary number is a multiple of \(i\), defined as the square root of negative one: \(i = \sqrt{-1}\). This concept is crucial when understanding how to handle terms like \(-3i^2\) in our calculation.
Remember, when \(i\) is squared, it turns into \(-1\):

• The equation \(i^2 = -1\) transforms expressions into real numbers.
• For example, \(-3i^2\) simplifies to \(-3(-1) = 3\).

This is a key step in our problem, turning potential complex calculations back into manageable terms.
Understanding this makes calculations with complex numbers, like our original expression
(3-i)(2+3i),
more straightforward. Here, \(-3i^2\) simplifies into +3,
illustrating how imaginary numbers primarily aid in these transformations.