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Chapter 7: Problem 32

Find each product. Express each answer in the form \(a+b i\) $$(4-i)(4+i)$$

Short Answer

Expert verified
The product is 17.

Step by step solution

01

Identify the binomials

Recognize that the expression \( (4 - i)(4 + i) \) involves two binomials.
02

Apply the FOIL method

Use the FOIL (First, Outer, Inner, Last) method to multiply the binomials. Calculate each multiplication separately: \[ \text{First: } 4 \times 4 = 16 \] \[ \text{Outer: } 4 \times i = 4i \] \[ \text{Inner: } (-i) \times 4 = -4i \] \[ \text{Last: } (-i) \times i = -i^2 \]
03

Combine like terms

Add the results of the FOIL method together: \[ 16 + 4i - 4i - i^2 \]
04

Simplify using the properties of imaginary numbers

Combine the like terms and simplify using \( i^2 = -1 \). The expression becomes: \[ 16 + 0i - (-1) = 16 + 1 = 17 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

FOIL method
The FOIL method is a handy technique for multiplying two binomials. It stands for First, Outer, Inner, and Last. This method helps to ensure that every term in the first binomial is multiplied by every term in the second.
Here is a breakdown of how it works:
  • First: Multiply the first terms in each binomial.
  • Outer: Multiply the outermost terms in each binomial.
  • Inner: Multiply the innermost terms in each binomial.
  • Last: Multiply the last terms in each binomial.
Let's apply it to the example: \( (4 - i)(4 + i) \):
\text{First: } 4 \times 4 = 16
\text{Outer: } 4 \times i = 4i
\text{Inner: } (-i) \times 4 = -4i
\text{Last: } (-i) \times i = -i^2. When you add these products together, you get: \ 16 + 4i - 4i - i^2 \. Notice that the middle terms, \(4i \) and \(-4i\) cancel each other out.
Imaginary Numbers
Imaginary numbers might sound complex, but they're simpler than you think! The imaginary unit is \(i\), defined as the square root of -1:
\( i = \sqrt{-1}\). This means that \(i^2 = -1\).
They extend our regular number system so that we can solve equations that don't have real solutions. For instance, there's no real number that satisfies \(x^2 = -1\), but \(i\) does because \(i^2 = -1\).
In the context of our exercise, when we see \(-i^2\), we can replace \(i^2\) with -1, simplifying our expression. So, \(-i^2\) becomes -(-1), which is just +1. This simplification helps us get to our final answer.
Binomials
A binomial is a polynomial with exactly two terms. They usually look like \(a + b\) or \(a - b\). In our example, the binomials are \(4 - i\) and \(4 + i\).
To multiply these binomials, we can use the FOIL method.
Binomials are useful in algebra because they can be squared, expanded, and factored in many interesting ways. When you multiply binomials, keep an eye on the terms you produce. Some of them may cancel out or combine, simplifying your work.
Our specific example ends up being simpler after using the properties of imaginary numbers, providing a great demonstration of how these concepts work together.

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