Problem 59 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Recognize the Components

Identify the components from each complex number in the expression \((3 - 2i)(2 + 3i)\). The first complex number is \(3 - 2i\) and the second complex number is \(2 + 3i\).

Step 2: Apply the Distributive Property

Multiply each term in the first complex number by each term in the second complex number, using the distributive property:\((3)(2) + (3)(3i) + (-2i)(2) + (-2i)(3i)\).

Step 3: Simplify Each Term

Calculate the product of each pair of terms:- \((3)(2) = 6\)- \((3)(3i) = 9i\)- \((-2i)(2) = -4i\)- \((-2i)(3i) = -6i^2\)

Step 4: Simplify Using \(i^2 = -1\)

Replace \(i^2\) with \(-1\) in \(-6i^2\): \(-6i^2 = -6(-1) = 6\).

Step 5: Combine Real and Imaginary Parts

Add all the real parts together:\(6 + 6 = 12\), and combine the imaginary parts:\(9i - 4i = 5i\).

Step 6: Write the Final Answer

Combine the results from the previous steps:The expression \((3 - 2i)(2 + 3i)\) simplifies to \(12 + 5i\).

Key Concepts

Distributive Property

The distributive property is a foundational principle in algebra that helps us expand expressions, especially when multiplying two binomials. It allows us to break down a complex multiplication into more manageable parts.
For complex numbers, this involves multiplying each term in one complex number by each term in another. It's like distributing each element of one set onto another.
This method guarantees we've considered all possible interactions between terms.

• Given two complex numbers: \(3 - 2i\) and \(2 + 3i\)
• Each term in \(3 - 2i\) is multiplied by each term in \(2 + 3i\)
• Results in: \(3 \cdot 2 + 3 \cdot 3i + (-2i) \cdot 2 + (-2i) \cdot 3i\)

Using this property always ensures you accurately distribute and account for every term.

Imaginary Unit

The imaginary unit, denoted as \(i\), is a mathematical concept used to extend the realm of real numbers. The defining property of the imaginary unit is \(i^2 = -1\).
This idea arises from the equation \(x^2 + 1 = 0\), where no real number satisfies the equation, thus introducing \(i\) to represent these values.
The presence of \(i\) allows us to describe numbers that are combinations of real and imaginary parts, also known as complex numbers.

• Example: \((-2i)\) indicates an imaginary component in a number.
• Special case: \((-2i)(3i) = -6i^2\).
• Substitute \(i^2\) with \(-1\): \-6(-1) = 6\.

An understanding of the imaginary unit aids in simplifying expressions involving complex numbers.

Multiplication of Complex Numbers

Multiplication of complex numbers involves using the distributive property combined with the properties of the imaginary unit. This process is slightly different than multiplying regular numbers due to the presence of both real and imaginary parts.

• Given two complex numbers like \(3 - 2i\) and \(2 + 3i\)
• Apply the distributive property: \(3 \cdot 2 + 3 \cdot 3i + (-2i) \cdot 2 + (-2i) \cdot 3i\)
• Simplify each term: \(6, 9i, -4i, -6i^2\)

When you multiply complex numbers, always calculate both the product of the real parts and the product of the imaginary parts, making sure to take into account \(i^2 = -1\). This ensures you not only handle the multiplication correctly but also simplify it fully.

Simplification in Algebra

Simplification in algebra typically involves combining like terms, using identities, and reducing expression complexity. When dealing with complex numbers, simplification ensures clarity and correctness, ultimately providing the answer in its simplest and most interpretable form.

• Combine like terms: interpret real and imaginary parts separately
• For example: from \(6 + 6 = 12\) and \(9i - 4i = 5i\)
• Simplified result: combine to form \(12 + 5i\)

Proper simplification reveals the result in the standard form \(a + bi\), essential for understanding and interpreting complex numbers in algebra and beyond.