Problem 61 - Intermediate Algebra

Question

Multiply. Write all answers in the form a \(+b i.\) $$ (4+\sqrt{-1})(3-\sqrt{-1}) $$

Step-by-Step Solution

Answer

The product is \( 13 - i \).

Step 1: Recognize the Imaginary Unit

Identify that \( \sqrt{-1} \) is \( i \), the imaginary unit. Rewrite each complex number using \( i \): \( (4 + i) \) and \( (3 - i) \).

Step 2: Distribute the Terms

Apply the distributive property, also known as the FOIL method, to multiply the expressions: \( (4+i)(3-i) = 4(3) + 4(-i) + i(3) + i(-i) \).

Step 3: Calculate Each Term

Calculate each component of the expanded expression: - \( 4 \times 3 = 12 \)- \( 4 \times (-i) = -4i \)- \( i \times 3 = 3i \)- \( i \times (-i) = -i^2 \).

Step 4: Substitute \( i^2 = -1 \) and Simplify

Since \( i^2 = -1 \), replace \( -i^2 \) with \(-(-1) = 1 \). Combine real and imaginary parts:- Real parts: \( 12 + 1 = 13 \)- Imaginary parts: \(-4i + 3i = -i \).

Step 5: Write the Final Answer

Combine the simplified real and imaginary parts into the final answer in the form \( a + bi \): \( 13 - i \).

Key Concepts

Imaginary Unit

In the world of complex numbers, the imaginary unit plays a crucial role. Often represented by the symbol \( i \), it is defined as the square root of negative one.
This means that \( i^2 = -1 \). Without \( i \), it would be impossible to deal with the square roots of negative numbers in mathematics, as they do not fit within the real number system. This clever concept allows us to expand beyond what is possible with just real numbers and solve equations that would otherwise have no solution.
By considering \( \ \sqrt{-1} \) as \( i \), we can rewrite expressions like \( 4 + \ \sqrt{-1} \) as \( 4 + i \), making calculations more manageable when dealing with complex numbers.

FOIL Method

The FOIL method is a specific technique used to multiply two binomials. The acronym FOIL stands for First, Outer, Inner, Last, referring to the order in which you multiply the terms.
To apply this method to our problem, which is to multiply \( (4 + i)(3 - i) \), we break it down as follows:

• First: Multiply the first terms: \( 4 \times 3 = 12 \)
• Outer: Multiply the outer terms: \( 4 \times (-i) = -4i \)
• Inner: Multiply the inner terms: \( i \times 3 = 3i \)
• Last: Multiply the last terms: \( i \times (-i) = -i^2 \)

After performing these calculations, the result is a sum from these products.
Using the FOIL method ensures a systematic approach to multiplying binomials so that no term is overlooked.

Distributive Property

The distributive property is fundamental in algebra and underlies the FOIL method. It states that multiplying a number by a sum is the same as doing each multiplication separately and then adding the results.
In mathematical terms, if \( a, b, \) and \( c \) are any numbers, the distributive property says \( a(b + c) = ab + ac \).

Applying this to complex numbers, when we distribute \( (4 + i) \) with \( (3 - i) \), we individually multiply each term:

• \( 4(3 - i) = 12 - 4i \)
• \( i(3 - i) = 3i - i^2 \)

By using the distributive property, we ensure that each term in \( (4+i) \) is multiplied with each term in \( (3-i) \).
This is key to properly expanding and simplifying expressions involving complex numbers.

Simplifying Expressions

Simplifying expressions, especially with complex numbers, requires combining like terms and understanding the properties of \( i \).
Each term produced from the multiplication step can be categorized into real or imaginary components:

• The real component is obtained from multiplying numbers without \( i \), and involves computation with \( i^2 \) since \( i^2 = -1 \), transforming it into a real number.
• The imaginary component involves terms with \( i \), and once multiplied or added, these terms are kept separate but alongside real parts to finalize the complex number form.

For the example \( (4+i)(3-i) \), simplifying means ensuring all real numbers and imaginary numbers are combined correctly to arrive at the equation:
The equation \( 13 - i \) illustrates the form \( a + bi \), which is the standard way to present a complex number, showing the real part \( a \) and the imaginary part \( bi \).
This systematic approach ensures clarity and correctness when handling complex numbers in expressions.