Problem 69 - A Graphical Approach to College Algebra

Question

Multiply as indicated. Write each product in standard form. $$3 i(2-i)^{2}$$

Step-by-Step Solution

Answer

The product in standard form is \(12 + 9i\).

Step 1: Expand the Exponent

First, let's expand the square of the binomial \((2-i)^2\). This can be done by using the formula for squaring a binomial, which is \((a-b)^2 = a^2 - 2ab + b^2\).For \(a = 2\) and \(b = i\), we have:\[(2-i)^2 = 2^2 - 2 \cdot 2 \cdot i + i^2\] \[= 4 - 4i + i^2\]Since \(i^2 = -1\), substitute \(i^2\) with \(-1\):\[= 4 - 4i - 1\]Simplify the expression:\[= 3 - 4i\]

Step 2: Multiply by the Scalar

Now, we need to multiply the result from Step 1 by \(3i\), as indicated in the original expression. Thus, we are multiplying:\[3i(3 - 4i)\]Distribute \(3i\) across each term inside the parentheses:\(3i \times 3 = 9i\)\(3i \times (- 4i) = -12i^2\)Since \(i^2 = -1\), this becomes:\(-12i^2 = -12(-1) = 12\)Thus, the product is:\[= 12 + 9i\]

Step 3: Write in Standard Form

The standard form for complex numbers is \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.From Step 2, we have:\[12 + 9i\]This is already in standard form with the real part \(12\) and the imaginary part \(9i\).

Key Concepts

Binomial Expansion

The Binomial Expansion is a powerful mathematical tool, allowing us to expand expressions raised to a power. It is super helpful in solving polynomial equations. In our exercise, we focused on squaring a binomial, which involves using the formula

• \((a-b)^2 = a^2 - 2ab + b^2\)

For the binomial \((2-i)^2\), we substitute

• \(a = 2\)
• \(b = i\)

This results in:

• \((2-i)^2 = 2^2 - 2 imes 2 imes i + i^2\)

By simplifying, you get:

• \(= 4 - 4i + i^2\)

Recognizing that \(i^2 = -1\) simplifies it further, thus giving:

• \(= 4 - 4i - 1 = 3 - 4i\)

This expansion is a vital skill in dealing with complex numbers.

Standard Form

Writing complex numbers in Standard Form, \(a + bi\), is essential for easy readability and comparison.
This form has two parts:

• \(a\) is the real part.
• \(bi\) is the imaginary part.

In our solution, after expanding and multiplying, we obtained:

• \(= 12 + 9i\)

Here:

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

Arranging in this form ensures that the complex number is clearly structured, aiding in further calculations or graphing in the complex plane.

Imaginary Unit

The Imaginary Unit, represented as \(i\), is a fundamental concept in complex numbers. It is defined with the property:

• \(i^2 = -1\)

This unique property allows us to work with numbers that extend beyond the real number line.
For example, in our exercise, when expanding and simplifying \((2-i)^2\), the term \(i^2 = -1\) allowed us to simplify the expression.
When multiplying with \(3i\), using \(i^2 = -1\) turned \(-12i^2\) into \(12\).
This transformation is crucial in finding the solution in a form that is easy to understand and work with further.
Understanding \(i\) is key to mastering complex numbers, as it forms the foundation of many advanced topics.