Problem 65 - Precalculus Student Solutions Manual 5th

Question

Find each product. Write the answer in standard form. $$3 i(2-i)^{2}$$

Step-by-Step Solution

Answer

12 + 9i

Step 1: Expand the Squared Term

Start by expanding the squared term \(2 - i\)^{2}. Use the formula for the square of a binomial: \(a - b\)^{2} = a^{2} - 2ab + b^{2}. In this case, \(a = 2\) and \(b = i\).\[ (2 - i)^{2} = 2^{2} - 2(2)(i) + i^{2} \]\[ = 4 - 4i + i^{2} \]

Step 2: Simplify Using Properties of Imaginary Numbers

Recall that \(i^{2} = -1\). Simplify the expression by substituting \(i^{2}\) with \ -1 \:\[ 4 - 4i + (-1) \]\[ = 4 - 4i - 1 \]\[ = 3 - 4i \]

Step 3: Multiply by 3i

Now multiply the result by 3i:\[ 3i(3 - 4i) \]Use the distributive property to expand:\[ = 3i \times 3 - 3i \times 4i \]\[ = 9i - 12i^{2} \]

Step 4: Simplify the Expression

Simplify the expression by substituting \i^{2} = -1\ again:\[ 9i - 12(-1) \]\[ = 9i + 12 \]Finally, express the answer in standard form (a + bi):\[ 12 + 9i \]

Key Concepts

Imaginary Numbers

Imaginary numbers are numbers that can be written as a real number multiplied by the imaginary unit, denoted as \(i\). The imaginary unit \(i\) is defined by the property that \(i^2 = -1\). This property is fundamental in complex number arithmetic and has important implications in various fields, including engineering and physics.

Some essential points about imaginary numbers include:

• The imaginary unit \(i\) is not a real number but can be combined with real numbers to form complex numbers.
• Any real number multiplied by \(i\) remains an imaginary number. For instance, \(3i\) and \(-5i\) are both imaginary numbers.
• Imaginary numbers are typically written in the form \(bi\), where \(b\) is a real number.

Understanding imaginary numbers is crucial because they form the foundation of complex numbers, which expand the concept of number systems beyond the real numbers.

Binomial Expansion

The binomial expansion is a method used to expand expressions that involve binomials raised to a power. A binomial is an algebraic expression with two terms, such as \(a - b\) or \(x + y\). In the given problem, we used the binomial expansion to expand \((2 - i)^2\).

To expand \((a - b)^2\), you can use the formula: \(a^2 - 2ab + b^2\). Applying this to \(2 - i\), where \(a = 2\) and \(b = i\), we get:

• \(2^2 = 4\)
• \(-2 \times 2 \times i = -4i\)
• \(i^2 = -1\)

Consequently, we have:
\((2 - i)^2 = 4 - 4i + (-1) = 3 - 4i\).

By understanding and using binomial expansion, you can simplify complex algebraic expressions and solve various mathematical problems.

Standard Form of Complex Numbers

The standard form of complex numbers is written as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. This format allows us to clearly identify and work with the real and imaginary components of complex numbers.

For example, in the expression \(12 + 9i\) from the problem:

• \(12\) is the real part.
• \(9i\) is the imaginary part.

Writing complex numbers in standard form makes arithmetic operations like addition, subtraction, multiplication, and division more manageable.
Here are some useful steps for ensuring that an expression is in standard form:

• Simplify the real and imaginary parts separately.
• Combine like terms.
• Make sure the imaginary unit \(i\) is appropriately placed in the term.

Using the standard form, complex numbers can be easily manipulated and understood, which is particularly valuable in higher mathematics and applied fields.