Problem 19 - College Algebra Essentials

Question

In Exercises \(9-20,\) find each product and write the result in standard form. $$(2+3 i)^{2}$$

Step-by-Step Solution

Answer

The product in standard form is \(-5 + 12i\).

Step 1: Write out the expression

First, write out the expression from the parentheses: \((2+3i)(2+3i)\)

Step 2: Apply FOIL Method

Then, apply the FOIL (First, Outside, Inside, Last) Method to compute the product: \(2*2 + 2*3i + 3i*2 + 3i*3i\). This simplifies to \(4 + 6i + 6i + 9i^{2}\)

Step 3: Factor in the property of i

Recall that \(i^{2} = -1\). Replace \(i^{2}\) with \(-1\): \(4 + 12i -9\).

Step 4: Simplify the expression

Finally, simplify the expression by combining like terms: \(-5 + 12i\)

Key Concepts

Imaginary Unit

The imaginary unit is a fundamental concept in the realm of complex numbers. It is represented by the symbol \(i\), and it is defined as the square root of -1. This may sound strange at first since there is no real number whose square is negative. However, \(i\) allows us to extend the real number system to include complex numbers.

When dealing with calculations involving \(i\), it's crucial to remember this property: \(i^2 = -1\).

• The imaginary unit is the basis for all complex numbers, which are numbers in the form of \(a + bi\), where \(a\) and \(b\) are real numbers.
• In this form, \(a\) is the real part, and \(bi\) is the imaginary part.
• The term imaginary doesn't convey that these numbers don't exist but rather that they extend our numerical understanding beyond real numbers.

In the expression \((2+3i)^2\), the value \(i^2\) becomes important because when calculated, it converts to \(-1\), thereby simplifying the expression.

FOIL Method

The FOIL method is a handy mnemonic and technique for expanding the product of two binomials in the form of \((a + b)(c + d)\). FOIL stands for First, Outside, Inside, Last, which indicates the order in which you should multiply the terms.

Here’s how it works step-by-step:

• First: Multiply the first terms in each binomial.
• Outside: Multiply the outer terms.
• Inside: Multiply the inner terms.
• Last: Multiply the last terms in each binomial.

Let's apply FOIL to the example \((2+3i)^2 = (2+3i)(2+3i)\):

• First: \(2 \times 2 = 4\)
• Outside: \(2 \times 3i = 6i\)
• Inside: \(3i \times 2 = 6i\)
• Last: \(3i \times 3i = 9i^2\)

After using FOIL, you combine the resultant elements: \(4 + 6i + 6i + 9i^2\). Remember, \(i^2 = -1\), which allows further simplification.

Standard Form

In mathematics, expressing numbers in an understandable format is crucial. For complex numbers, the standard form is \(a + bi\). Here, \(a\) represents the real number component, and \(bi\) is the imaginary component.

It's essential to combine like terms to achieve this format. Returning to our calculated expression, \(4 + 12i - 9\), we simplify it by gathering the real numbers and the imaginary numbers to present it in standard form:

• Combine real numbers: \(4 - 9 = -5\).
• Combine imaginary numbers: The terms involving \(i\) are already consolidated as \(12i\).

Therefore, the standard form of the original expression \((2+3i)^2\) is \(-5 + 12i\). This clear form gives both the real and imaginary components distinctly, helping to easily recognize and interpret complex numbers.