Problem 17 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The product of the two complex numbers \((-5+i)\) and \((-5-i)\) in standard form is 24.

Step 1: Apply distributive property

Use the distributive property (also known as the FOIL method, which stands for 'First, Outer, Inner, Last') to multiply the two complex numbers together. This results in: \[(-5+i)(-5-i)=(-5*-5)+(-5*-i)+(i*-5)+(i*-i)\]

Step 2: Simplify the result

Next, simplify the result. Remember, \(i^2\) equals -1. So this leads to:\[((-5*-5)+(-5*-i)+(i*-5)+(i*-i))=(25+5i-5i-1)=24\]

Step 3: Write in standard form

Lastly, the resultant number is a real number, so it's standard form is the number itself, which is 24.

Key Concepts

Distributive Property

The distributive property is a fundamental principle used in multiplication. It's like a mathematical way to handle multiplying a number by a group of numbers added together. Think of it like distributing a set of tasks among a team. When it comes to complex numbers, you can apply the distributive property to multiply the parts separately and then add them together.
For example, when given

• The expression \((-5+i)(-5-i)\), you treat it like \((a+b)(c+d)\),
• where you distribute each term in the first parenthesis to each term in the second parenthesis,
• this results in multiplying 'First', 'Outer', 'Inner', and 'Last' terms separately.

The distributive property ensures each component is accounted for in the multiplication, paving the way for further simplification.

FOIL Method

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

• First: Multiply the first terms in each binomial together.
• Outer: Multiply the outer terms in the product.
• Inner: Multiply the inner terms.
• Last: Multiply the last terms.

When dealing with the given expression \((-5+i)(-5-i)\):

• First: \(-5 \times -5 = 25\)
• Outer: \(-5 \times -i = 5i\)
• Inner: \(i \times -5 = -5i\)
• Last: \(i \times -i = -1\)

By following these steps, you combine the results: \(25 + 5i - 5i - 1\).
Notice how the imaginary parts \(5i\) and \(-5i\) cancel each other out, simplifying to \(25 - 1 = 24\). This showcases the effectiveness of the FOIL method in handling multiplication of binomials.

Imaginary Unit

The imaginary unit is a key component in the realm of complex numbers. Represented by the symbol \(i\), it has a unique property: \(i^2 = -1\). This concept allows us to extend the real number system.

In calculations, the imaginary unit helps resolve expressions that involve square roots of negative numbers. For instance, in the product \(i \times -i\), the use of \(i^2 = -1\) is crucial. It simplifies the expression \(i \times -i\) to \(-1\).

Understanding the imaginary unit helps in mastering operations with complex numbers. It acts as a building block for forming complex solutions, easing the process of solving equations that don't resolve neatly in the real number world.