Problem 75 - Algebra for College Students

Question

Find each of the products and express the answers in the standard form of a complex number. $$(9+6 i)(-1-i)$$

Step-by-Step Solution

Answer

The product is \(-3 - 15i\).

Step 1: Set Up the Multiplication

Consider the two complex numbers \( (9 + 6i) \) and \( (-1 - i) \). To find the product, we need to multiply these two expressions using the distributive property (FOIL method).

Step 2: Apply the Distributive Property

Apply the distributive property to expand the expression: \[(9 + 6i)(-1 - i) = 9(-1) + 9(-i) + 6i(-1) + 6i(-i)\]

Step 3: Simplify Each Term

Calculate each term separately: - \(9(-1) = -9\)- \(9(-i) = -9i\)- \(6i(-1) = -6i\)- \(6i(-i) = -6i^2\)

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

Since \(i^2 = -1\), the term \(-6i^2\) becomes \(-6(-1)\), which is \(6\).

Step 5: Combine Like Terms

Combine the real and imaginary parts together:- Real part: \(-9 + 6 = -3\)- Imaginary part: \(-9i - 6i = -15i\)Thus, the expression simplifies to \(-3 - 15i\).

Key Concepts

FOIL Method

The FOIL Method stands for First, Outer, Inner, Last. It is a technique commonly used in algebra to multiply two binomials. When given an expression with two complex numbers, such as \((a + bi)(c + di)\), the FOIL method is used to expand the product. This is done by:

• First: Multiply the first terms of each binomial: \(a \cdot c\)
• Outer: Multiply the outer terms: \(a \cdot di\)
• Inner: Multiply the inner terms: \(bi \cdot c\)
• Last: Multiply the last terms: \(bi \cdot di\)

Together, these match the distributive steps you might already know. The FOIL method is an efficient way to ensure all parts of the products are considered. It is particularly helpful when working with complex numbers, as it naturally leads to combining like terms and simplifies expressions into their standard forms.

Distributive Property

The distributive property is a key algebraic property used for expansion in multiplication. When dealing with the multiplication of complex numbers, such as \((9 + 6i)(-1 - i)\), the distributive property ensures each term in the first binomial is multiplied by each term in the second binomial. This results in:

• \(9 \times -1 = -9\)
• \(9 \times -i = -9i\)
• \(6i \times -1 = -6i\)
• \(6i \times -i = -6i^2\)

The beauty of the distributive property lies in its simplicity; it makes sure that no terms are left out during multiplication. This is essential when working with complex numbers because it allows the expression to be simplified correctly by combining like terms after applying this property.

Standard Form of a Complex Number

The standard form of a complex number is \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. After performing operations with complex numbers, like multiplication, it's important to express the result in this form.
In our example, multiplying \((9 + 6i)(-1 - i)\) gives \(-3 - 15i\). Here:

• The real part is \(-3\)
• The imaginary part is \(-15i\)

The process involves calculating each term, combining like terms, and ensuring the result is neatly expressed in \(a + bi\) form. This clear expression helps in understanding and graphically visualizing complex numbers.

Imaginary Unit

The imaginary unit, denoted as \(i\), is the foundation of complex numbers. By definition, \(i^2 = -1\). This property is crucial when performing arithmetic with complex numbers.
During multiplication, terms like \(6i(-i)\) appear, which translate into \(-6i^2\). Recognizing that \(i^2 = -1\) transforms \(-6i^2\) into \(6\). Without understanding the imaginary unit, simplifying such expressions would be impossible. By mastering the imaginary unit, you'll unlock the ability to work effortlessly with complex numbers, enabling you to solve problems involving deeper algebraic concepts.