Problem 15 - College Algebra

Question

In Exercises \(9-20,\) find each product and write the result in standard form. $$(3+5 i)(3-5 i)$$

Step-by-Step Solution

Answer

The product of the complex numbers \(3+5i\) and \(3-5i\) is 34.

Step 1: Define the Complex Numbers

The complex numbers in this exercise are \(3+5i\) and \(3-5i\). These are conjugates of each other, as they have the same real part and opposite imaginary parts.

Step 2: Multiply the Complex Numbers

To find a product of these numbers, use the distributive property of multiplication over addition, also known as the FOIL method. This stands for First (multiply the first terms in each binomial), Outer (multiply the outer terms), Inner (multiply the inner terms), and Last (multiply the last terms). This leads to: \((3+5i)(3-5i) = 3*3 + 3*(-5i) +5i*3 -5i*5i\).

Step 3: Simplify the expression

Rewrite the expression \((3+5i)(3-5i) = 9 -15i +15i -25i^2\). Recall that \(i^2\) equals to -1. So this simplifies to \(9 + 25 = 34\).

Key Concepts

Conjugate Complex Numbers

Conjugate complex numbers are a pair of complex numbers that have the same real part but opposite signs for their imaginary parts. In our example, we have the complex numbers \(3 + 5i\) and \(3 - 5i\). These are conjugates of each other. Conjugate complex numbers have a fascinating property when multiplied together: their imaginary parts cancel, leaving a real number. Here’s what happens:

• The product of conjugate complex numbers will always be a real number because the imaginary components subtract away due to opposite signs.
• The result is essentially the square of the real part plus the square of the imaginary part.

Understanding conjugate complex numbers is essential because they help simplify expressions and solve quadratic equations that involve complex roots. Using conjugates also aids in finding the modulus of a complex number.

Distributive Property

The distributive property is a fundamental principle for the multiplication of numbers, which means that \(a(b + c) = ab + ac\). This property is crucial when dealing with expressions that include sums and products, such as multiplying complex numbers.In our example, we applied the distributive property to expand \((3 + 5i)(3 - 5i)\):

• First term: \(3 \times 3 = 9\)
• Outer term: \(3 \times (-5i) = -15i\)
• Inner term: \(5i \times 3 = 15i\)
• Last term: \(5i \times (-5i) = -25i^2\)

The distributive property allows you to break down and manage even the most complex expressions by multiplying and then combining like terms. This approach is very effective in making calculations more manageable and clear.

FOIL Method

The FOIL method is a particular application of the distributive property for multiplying two binomials. FOIL stands for:

• First: Multiply the first terms
• Outer: Multiply the outer terms
• Inner: Multiply the inner terms
• Last: Multiply the last terms

In our exercise, multiplying the binomials \((3 + 5i)\) and \((3 - 5i)\) involves using the FOIL method to ensure each part of the binomial is multiplied properly:

• The First is \(3 \times 3 = 9\).
• The Outer is \(3 \times (-5i) = -15i\).
• The Inner is \(5i \times 3 = 15i\).
• The Last is \(5i \times (-5i) = -25i^2\).

After applying the FOIL method, you simplify the expression by combining like terms.

Imaginary Unit

The imaginary unit, denoted as \(i\), is a mathematical concept used to extend the concept of square roots to negative numbers. It is defined by the equation \(i^2 = -1\). The imaginary unit is fundamental when working with complex numbers, as it represents the imaginary part.In our calculation, the term \(-25i^2\) simplifies using the property of the imaginary unit, where \(i^2 = -1\). So, \(-25i^2\) becomes \(-25(-1) = 25\). This transformation is key to converting complex products into a real number.Understanding the role of the imaginary unit is essential for performing arithmetic operations with complex numbers and simplifying expressions.