Problem 69 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

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

Step 1: Expand the Binomial

First, expand the expression \((2 - i)^2\) using the formula for the square of a binomial: \((a - b)^2 = a^2 - 2ab + b^2\). Here, \(a = 2\) and \(b = i\).Thus, \((2 - i)^2 = 2^2 - 2 \times 2 \times i + i^2 = 4 - 4i + i^2\).

Step 2: Simplify the Imaginary Unit

Recall that \(i^2 = -1\). Substitute this into the expression:\(4 - 4i + i^2 = 4 - 4i - 1\). This simplifies to: \(3 - 4i\).

Step 3: Distribute the Imaginary Number

Multiply \(3i\) by each term in the expression \(3 - 4i\):\(3i(3) = 9i\), and\(3i(-4i) = -12i^2\).

Step 4: Simplify the Expression

Combine the results from the previous step. Since \(i^2 = -1\), replace \(-12i^2\) with \(-12(-1) = 12\):Thus, the expression becomes: \(9i + 12\).Reorder the terms for standard form: \(12 + 9i\).

Key Concepts

Imaginary Unit

In mathematics, the imaginary unit is denoted as \(i\) and is defined by the property \(i^2 = -1\). This mathematical construct extends the real number system to the complex number system by introducing a number whose square is negative. While real numbers can only handle non-negative squares, imaginary numbers allow us to solve equations such as \(x^2 + 1 = 0\). The imaginary unit is foundational for complex numbers, where numbers are expressed in the form \(a + bi\), with \(a\) and \(b\) being real numbers.

• When squared, the imaginary unit results in a negative real number, adhering to \(i^2 = -1\).
• Imaginary numbers are used in diverse fields such as engineering, physics, and signal processing.

Understanding this concept is crucial when simplifying expressions involving \(i\) and applying it during calculations in complex arithmetic.

Standard Form

The standard form of a complex number is expressed as \(a + bi\) where \(a\) is the real part, and \(bi\) is the imaginary part. It provides a clear way to write complex numbers, allowing for easy identification of their real and imaginary components.

• Typically, the real part \(a\) is written first, followed by the imaginary part \(bi\).
• For example, after simplifying the expression \(3i(2-i)^2\), the result is written in standard form as \(12 + 9i\).

Writing numbers in standard form is essential when performing arithmetic operations on complex numbers. This ensures clarity and avoids confusion in calculations.

Binomial Expansion

Binomial expansion involves expanding expressions that are raised to a power, particularly those involving binomials like \((a + b)^n\). When dealing with complex expressions, the expansion formula \((a - b)^2 = a^2 - 2ab + b^2\) is extremely useful for turning a binomial expression into a trinomial.

Example Application

When given \((2 - i)^2\), we identify \(a = 2\) and \(b = i\). Applying the formula results in:\[(2 - i)^2 = 2^2 - 2 \times 2 \times i + i^2\]This simplifies to \(4 - 4i + (-1)\), which further reduces to \(3 - 4i\) as \(i^2 = -1\).Understanding binomial expansion is key to breaking down and simplifying polynomial expressions, especially those involving complex numbers.

Polynomial Multiplication

Polynomial multiplication involves distributing terms in one polynomial over terms in another, often leading to the combination of like terms. It requires careful application of the distributive property to ensure all parts of the expression are accounted for.

Complex Number Context

When multiplying a complex number by a polynomial, each term must be multiplied separately. For instance, with the product \(3i(3 - 4i)\), we perform:

• \(3i \cdot 3 = 9i\)
• \(3i \cdot (-4i) = -12i^2 = 12\) (since \(i^2 = -1\))

The resulting expression \(9i + 12\) is then reordered to the standard form \(12 + 9i\).Polynomials involving complex numbers often lead to results comprising both real and imaginary components, emphasizing the importance of proper arrangement and simplification.