Problem 26 - Algebra and Trigonometry

Question

Evaluate the expression and write the result in the form a bi. $$ (5-3 i)(1+i) $$

Step-by-Step Solution

Answer

The expression evaluates to \(8 + 2i\).

Step 1: Apply the Distributive Property

To evaluate the expression \((5 - 3i)(1 + i)\), use the distributive property (also known as the FOIL method) to multiply the two binomials. This means you'll multiply each term in the first binomial by each term in the second binomial:1. Multiply \(5\) by \(1\). 2. Multiply \(5\) by \(i\). 3. Multiply \(-3i\) by \(1\). 4. Multiply \(-3i\) by \(i\).

Step 2: Calculate Each Multiplication

Calculate each multiplication from the previous step:1. \(5 \times 1 = 5\).2. \(5 \times i = 5i\).3. \(-3i \times 1 = -3i\).4. \(-3i \times i = -3i^2\).

Step 3: Simplify the Complex Multiplication

Remember that \(i^2 = -1\). Thus, the expression \(-3i^2\) simplifies to \(-3(-1) = +3\).

Step 4: Combine All Terms

Combine all the terms derived from the multiplications:\(5 + 5i - 3i + 3\).

Step 5: Simplify the Expression

Combine like terms:- Real parts: \(5 + 3 = 8\).- Imaginary parts: \(5i - 3i = 2i\).The simplified form of the expression is \(8 + 2i\).

Key Concepts

Imaginary Unit

The imaginary unit, denoted as \(i\), is a fundamental concept in complex numbers. It is defined by the property \(i^2 = -1\). This definition is the cornerstone of operations involving complex numbers, which are numbers that have both a real part and an imaginary part.
Understanding the imaginary unit helps in simplifying expressions where \(i\) appears in multiplicative operations. For example, in the expression \(-3i^2\), knowing that \(i^2 = -1\) allows us to substitute \(-1\) for \(i^2\). This simplifies \(-3i^2\) to \(-3(-1) = 3\).
This conversion is crucial when performing arithmetic operations with complex numbers, as it transforms complex squares into real terms, simplifying the combination into the real part of the final result.

Distributive Property

The Distributive Property is a rule in algebra that allows you to multiply a single term and two or more terms inside a set of parentheses. It states that: - \(a(b + c) = ab + ac\).When applied to complex numbers, like in our expression \((5-3i)(1+i)\), the distributive property helps in expressing the multiplication of these two binomials as a series of simpler multiplications.
Here's how it works in the case of complex numbers:

• Multiply the real part of the first binomial by both parts of the second binomial.
• Then, multiply the imaginary part of the first binomial by both parts of the second binomial.

The result consists of four multiplications, combining them gives the expanded form, which can then be simplified further to form the complex number in \(a + bi\) format.

FOIL Method

The FOIL method is a specific application of the distributive property used to multiply two binomials. It stands for First, Outer, Inner, Last, which refers to the order in which you multiply the terms:

• First: Multiply the first terms of each binomial.
• Outer: Multiply the outer terms. This would be the first term of the first binomial and the second term of the second binomial.
• Inner: Multiply the inner terms. The last term of the first binomial and the first term of the second binomial.
• Last: Multiply the last terms of each binomial.

In doing so, for the expression \((5-3i)(1+i)\), you do:

• First: \(5 \times 1 = 5\)
• Outer: \(5 \times i = 5i\)
• Inner: \(-3i \times 1 = -3i\)
• Last: \(-3i \times i = -3i^2 = 3\) (using the fact that \(i^2 = -1\))

After applying FOIL, combine like terms to simplify: the real components (\(5 + 3 = 8\)) and the imaginary components (\(5i - 3i = 2i\)). The result is the complex number \(8 + 2i\). By practicing the FOIL method, students can gain a strong tool for handling and simplifying expressions involving complex numbers.