Problem 59 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The product is \(12 + 5i\).

Step 1: Apply the Distributive Property

To multiply the two complex numbers, use the distributive property (FOIL method, stands for First, Outer, Inner, Last). Begin with the first terms: multiply the real parts: \[ 3 \times 2 = 6 \] Next, multiply the outer terms: the real part of the first complex number and the imaginary part of the second:\[ 3 \times 3i = 9i \]

Step 2: Multiply Inner Terms

Now, multiply the inner terms: the imaginary part of the first complex number and the real part of the second:\[ -2i \times 2 = -4i \]

Step 3: Multiply the Last Terms and Simplify

Finally, multiply the imaginary parts of both complex numbers:\[ -2i \times 3i = -6i^2 \] Since \(i^2 = -1\), this becomes:\[ -6(-1) = 6 \]

Step 4: Combine Real and Imaginary Terms

Now combine all the results:Real parts: \(6 + 6 = 12\)Imaginary parts: \(9i - 4i = 5i\)Therefore, the product of the complex numbers is:\[ 12 + 5i \]

Key Concepts

Distributive Property

When we talk about the distributive property in mathematics, it is a fundamental concept that helps us understand how to break down expressions and simplify calculations. It works by distributing one quantity across each term inside a set of parentheses. For instance, if you have an expression like

• \(a(b + c)\)

this becomes:

• \(ab + ac\)

In the context of complex numbers, this property allows us to multiply each part of one complex number by each part of another. This is essential because complex numbers have both real and imaginary components. When multiplying

• \((3 - 2i)(2 + 3i)\)

it's the distributive property that ensures each term is accounted for individually. Thus, breaking it into four smaller multiplications:

• First, multiply the real parts.
• Next, multiply the cross terms (one real, one imaginary).
• Follow this by multiplying the inner imaginary and outer real parts.
• Lastly, multiply the imaginary parts.

This step-by-step process highlights how mathematics allows us to structure problems logically for easier understanding and solving.

FOIL Method

The FOIL method is a handy mnemonic to remember how to apply the distributive property when multiplying two binomials. FOIL stands for:

• First: Multiply the first terms of each binomial.
• Outer: Multiply the outer terms.
• Inner: Multiply the inner terms.
• Last: Multiply the last terms.

For complex numbers, this becomes especially helpful because each binomial like

• \((3-2i)(2+3i)\)

contains both a real and an imaginary part. By applying FOIL, we systematically break the multiplication into manageable parts. Bear in mind, when you multiply the last terms, you deal with imaginary units which yield special results since

• \(i^2 = -1\).

This step is key for identifying the real addition to our final result. Learning FOIL not only aids in complex multiplication but provides a universal approach for binomial expressions.

Real and Imaginary Parts

Complex numbers have a unique structure represented by two components: the real part and the imaginary part. They are typically written in the form

• \(a + bi\)

where \(a\) is the real part and \(bi\) is the imaginary part. This means every complex number exists on a plane, with the real component moving horizontally and the imaginary moving vertically. Understanding this helps in visualizing operations like addition, subtraction, and multiplication.
After multiplying complex numbers using previous concepts, combining these results into a single expression is crucial. Our solution for

• \((3-2i)(2+3i)\)

led us to two separate sums:

• Real: \(6 + 6\)
• Imaginary: \(9i - 4i\)

By adding these separately, we maintain clarity and properly align real with real and imaginary with imaginary parts. Thus, our final result

• \(12 + 5i\)

showcases both parts harmoniously. This focus secures our understanding of how to effectively manage and simplify such numbers, especially pivotal in various fields of math and science.