Problem 57 - Intermediate Algebra

Question

Multiply. Write all answers in the form a \(+b i.\) $$ (2+i)(3-i) $$

Step-by-Step Solution

Answer

The product is \(7 + i\).

Step 1: Apply the Distributive Property

Use the distributive property to expand the expression: \[(2+i)(3-i) = 2(3) + 2(-i) + i(3) + i(-i)\]

Step 2: Perform Multiplications

Multiply each pair of terms:\[2 \times 3 = 6\]\[2 \times (-i) = -2i\]\[i \times 3 = 3i\]\[i \times (-i) = -i^2\]

Step 3: Simplify Using \(i^2 = -1\)

Notice that \(i^2 = -1\), so simplify \(-i^2\) as follows:\[-i^2 = -(-1) = 1\]

Step 4: Combine Like Terms

Add together the real terms and the imaginary terms:Real: \(6 + 1 = 7\)Imaginary: \(-2i + 3i = 1i = i\)Hence, the expression simplifies to:\[7 + i\]

Key Concepts

Distributive Property

The distributive property is a key principle in algebra that helps simplify complex expressions. When you encounter expressions involving parentheses, the distributive property allows you to multiply each term within the parentheses by another term outside. This ensures all parts of the equation are accounted for.
For example, to solve \((2+i)(3-i)\), we use this property as follows:

• Multiply \(2\) by \(3-i\)
• Multiply \(i\) by \(3-i\)

Hence, it expands to \[2(3) + 2(-i) + i(3) + i(-i)\].
Each multiplication is carried out separately, allowing us to work methodically and reducing errors. It's like distributively giving a particular value to several others within a group or bracket. This principle is not just handy for dealing with simple integers or real numbers, but also invaluable when working with complex numbers.

Imaginary Unit

The imaginary unit, represented as \(i\), is foundational in the field of complex numbers. It is defined by the property \(i^2 = -1\). This concept extends the number system beyond real numbers to allow for the solution of equations like \(x^2 + 1 = 0\).
In this context, \(i\) acts much like any variable when considered under operations such as addition, subtraction, multiplication, and division, but with its unique transformation under squaring.
When you multiply two imaginary units, you apply \(i^2\) which results in \(-1\). This transformation is a crucial simplification during calculations involving complex numbers.
For instance, in our exercise, multiplying \(i\) and \(-i\) results in \[-i^2\], which simplifies further using the property of \(i^2\):

• \(-i^2 = -(-1) = 1\)

This step is essential for converting complex expressions into simpler, more manageable forms.

Simplifying Expressions

Simplifying expressions involving complex numbers often entails combining terms to reduce them to their simplest form, often a combination of a real and imaginary part like \(a + bi\).
Following the initial expansion using the distributive property, the next step is careful arithmetic. We multiply each term and use the properties of the imaginary unit to transform parts of the expression.
For example, after multiplying, we analyze:

• The real components: \(2 \times 3\) and \(-i^2\)
• The imaginary components: \(2 \times (-i)\) and \(i \times 3\)

Combining the like terms is pivotal:

• Add the real numbers: \(6 + 1\) to get \(7\)
• Add the imaginary numbers: \(-2i + 3i\) to arrive at \(1i = i\)

This process results in a simplified form of the complex expression, \(7 + i\), making it easier to understand and apply in further mathematical contexts.