Problem 60 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The product is \(9 + 7i.\)

Step 1: Distribute the First Term

First, distribute the first term of the first parenthesis, which is 3, across the second parenthesis. This will give us: \[3 imes (2 + 3i) = 3 imes 2 + 3 imes 3i = 6 + 9i. \]

Step 2: Distribute the Second Term

Next, distribute the second term of the first parenthesis, which is \(-i\), across the second parenthesis. This will give us: \[-i imes (2 + 3i) = -i imes 2 - i imes 3i = -2i - 3i^2. \]

Step 3: Simplify the Product

Combine all the terms obtained from the distribution steps: \[6 + 9i - 2i - 3i^2. \]Since \(i^2 = -1\), replace \(-3i^2\) with \(3\):\[6 + 9i - 2i + 3. \]

Step 4: Combine Like Terms

Now, combine the real parts and the imaginary parts: Real part: \[6 + 3 = 9.\]Imaginary part: \[9i - 2i = 7i.\]Thus, the expression simplifies to: \[9 + 7i.\]

Key Concepts

Distributive Property

Using the distributive property in algebra is an essential skill, particularly when working with expressions involving both numbers and variables. It's essentially a method for multiplying one term by a group of terms inside a parenthesis. The idea is to "distribute" the outer term to each component inside, step by step.

• Imagine we have an expression, say, \(a(b+c)\). In simpler terms, what we do is multiply each element inside the parenthesis by \(a\). So, \(a(b+c)\) becomes \(ab + ac\).
• This same rule applies when dealing with more complex numbers or expressions, such as \(3(2 + 3i)\), where we distribute the \(3\) to both \(2\) and \(3i\).

Applying this principle helps simplify an otherwise complicated multiplication process, making it a foundational technique for tackling various algebraic problems.

Imaginary Unit

The imaginary unit is a concept that might initially seem abstract but is remarkably useful in handling certain types of mathematical equations, particularly those involving square roots of negative numbers. Denoted typically as \(i\), the imaginary unit is defined by the property that \(i^2 = -1\).
This characterization opens up a new realm in math called complex numbers. A complex number takes the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) represents the imaginary part.

• This allows us to extend real number solutions to include answers that involve \(i\), essential when dealing with certain polynomial equations.
• For example, in the solution, multiplying \(-i(3i)\) involves using \(i^2 = -1\), which simplifies complex expressions significantly.

Understanding the imaginary unit provides a valuable tool for solving equations that are not possible with real numbers alone.

Polynomial Multiplication

Polynomial multiplication is an extension of the distributive property applied within the realm of polynomials. This process involves multiplying each term of one polynomial by each term of another, similar to how we dealt with the expression \( (3-i)(2 + 3i) \) in the original exercise.
Here's how it works using our example:

• Firstly, you multiply the first term of the first binomial (3) with both terms of the second binomial (2 and 3i).
• Secondly, you multiply the second term of the first binomial (-i) with both terms of the second binomial.

Ultimately, these steps are grounded in understanding how multiple terms interact through arithmetic and a bit of clever distribution.
By methodically applying polynomial multiplication, we bring complex expressions into a more manageable form, often combining like terms and simplifying further as required.