Problem 29 - Algebra 2 and Trigonometry

Question

In \(26-37,\) find each product. $$ (4-5 i)(3-2 i) $$

Step-by-Step Solution

Answer

The product is \(2 - 23i\).

Step 1: Apply the Distributive Property

To find the product of \( (4-5i)(3-2i) \), apply the distributive property (also known as the FOIL method for binomials). This method involves multiplying each term in the first binomial by each term in the second binomial:\[ \begin{align*}(4-5i)(3-2i) & = 4 imes 3 + 4 imes (-2i) + (-5i) imes 3 + (-5i) imes (-2i) \& = 12 - 8i - 15i + 10i^2\end{align*} \]

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

Replace \( i^2 \) with \(-1\) to simplify the expression. The term \( 10i^2 \) becomes \( 10(-1) = -10 \):\[ 12 - 8i - 15i - 10 \]

Step 3: Combine Like Terms

Combine the real parts and the imaginary parts separately:Real Parts: \( 12 - 10 = 2 \)Imaginary Parts: \( -8i - 15i = -23i \)Thus, the expression simplifies to:\[ 2 - 23i \]

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that allows us to multiply a single term by multiple terms inside a parenthesis. It can be expressed as:

• For any numbers or expressions, a, b, and c: \(a(b + c) = ab + ac\).

When two binomials are multiplied, like \((4 - 5i)(3 - 2i)\), each term of the first binomial is distributed across each term of the second binomial. That's why it results in four terms to evaluate:

• First: \(4 \times 3\)
• Outside: \(4 \times -2i\)
• Inside: \(-5i \times 3\)
• Last: \(-5i \times -2i\)

Understanding this property helps simplify expressions and solve equations, providing a solid foundation for delving into more complex algebraic concepts.

Imaginary Numbers

Imaginary numbers are a fascinating part of mathematics, emerging from the need to solve equations that don't have real solutions. The core idea stems from the unit imaginary number, denoted as \(i\), with the property that:

• \(i^2 = -1\).

In the solved problem \((4 - 5i)(3 - 2i)\), each appearance of \(i\) represents a multiplication with the imaginary unit. When multiplying imaginary numbers, it's crucial to remember that any \(i^2\) encountered will turn into \(-1\). This concept is key in simplifying expressions involving imaginary terms.
By understanding how to manage powers of \(i\), students can fully engage with complex number operations, making otherwise unsolvable equations approachable in the realm of complex numbers.

• Imaginary numbers play a critical role in advanced mathematics and physics by helping to address dimensions beyond the real number line.

FOIL Method

The FOIL method is a handy mnemonic for multiplying two binomials, specifically in the form \((a + b)(c + d)\). It stands for:

• First (\(a \times c\))
• Outside (\(a \times d\))
• Inside (\(b \times c\))
• Last (\(b \times d\))

In the expression \((4-5i)(3-2i)\), the FOIL method helps keep track of every interaction between the terms:

• \(First: 4 \times 3 = 12\)
• \(Outside: 4 \times -2i = -8i\)
• \(Inside: -5i \times 3 = -15i\)
• \(Last: -5i \times -2i = 10i^2\)

After applying FOIL, you simplify by combining like terms, and substituting \(i^2 = -1\). This approach ensures that your multiplication process is methodical and organized.
The real and imaginary parts are combined at the end to give the final expression, showing how practicing the FOIL method corrects intuition in complex number multiplication. Employing FOIL effectively makes complex problems much more manageable, aiding comprehension and accuracy in mathematics.