Problem 29 - Elementary Algebra

Question

Perform the operations. $$ (1-i)(8-9 i) $$

Step-by-Step Solution

Answer

The result of \((1-i)(8-9i)\) is \(-1 - 17i\).

Step 1: Distribute the Terms

To solve \((1-i)(8-9i)\), use the distributive property: Multiply each term in the first binomial by each term in the second binomial. This is sometimes referred to as FOIL (First, Outer, Inner, Last) for binomials.

Step 2: Multiply the First Terms

Multiply the first terms of each binomial.That is, multiply \(1\) and \(8\): \[1 \times 8 = 8\].

Step 3: Multiply the Outer Terms

Multiply the outer terms, which are \(1\) and \(-9i\): \[1 \times (-9i) = -9i\].

Step 4: Multiply the Inner Terms

Multiply the inner terms, which are \(-i\) and \(8\): \[-i \times 8 = -8i\].

Step 5: Multiply the Last Terms

Multiply the last terms of each binomial:Multiply \(-i\) and \(-9i\): \[(-i) \times (-9i) = 9i^2\]. Remember: Since \(i^2 = -1\), this becomes \[9(-1) = -9\].

Step 6: Add All the Results

Now, add up all the results from Steps 2 through 5:\[8 + (-9i) + (-8i) + (-9)\].Combine like terms: \[(8 - 9) + (-9i - 8i) = -1 - 17i\].

Key Concepts

Distributive Property

The distributive property is a helpful mathematical rule. It makes it possible to multiply a single term by each term within parentheses. For example, in the expression \((a+b)(c+d)\), the distributive property helps us spread out the multiplication. Here's how it breaks down:

• First, multiply \(a\) by \(c\) and by \(d\).
• Then, multiply \(b\) by \(c\) and by \(d\).

Put simply, each term in the first group gets its chance to interact with each term from the second group. The distributive property ensures that when dealing with expressions involving addition and multiplication, no part is left out.

Binomials

A binomial is an algebraic expression containing two distinct terms. Commonly, it looks like \(a + b\) or \(x - y\). This straightforward structure is often seen in problems requiring multiplication or simplification, as with the expression \((1-i)(8-9i)\) from our given problem.Binomials are flexible and foundational in algebra and complex numbers. They help us explore relationships between numbers, both real and imaginary. Working with binomials offers practice in using and understanding the essentials of algebraic operations.

FOIL Method

The FOIL method is a unique strategy used specifically for multiplying two binomials. It stands for "First, Outer, Inner, Last" and guides you through the steps to make sure no terms are overlooked. Let’s see how it works:

• First: Multiply the first terms of each binomial, such as \(1\) and \(8\).
• Outer: Multiply the outer terms, or \(1\) and \(-9i\).
• Inner: Multiply the inner terms, which are \(-i\) and \(8\).
• Last: Multiply the last terms, namely \(-i\) and \(-9i\).

Each step makes sure every part of the binomials is factored in. In algebra, especially with complex numbers, missing any part can lead to incorrect results.

Imaginary Unit

The imaginary unit, \(i\), is a fundamental concept in complex numbers, representing the square root of \(-1\). Although it might seem abstract, \(i\) allows mathematicians, scientists, and engineers to solve equations that traditional real numbers can't handle.When squaring \(i\), an interesting property emerges: \(i^2 = -1\). This fact is critical when multiplying complex numbers, as it transforms expressions involving powers of \(i\). For example, if we encounter \(9i^2\), we recognize this as \(9(-1)\), simplifying to \(-9\).Understanding and using \(i\) is essential for working with complex numbers; it's a stepping stone to more advanced concepts in mathematics and engineering. With \(i\), we venture into new realms that extend beyond the limitations of real numbers.