Problem 55 - Precalculus with Calculus Previews

Question

Factor the polynomial function \(f(x)=x^{2}-i\)

Step-by-Step Solution

Answer

The polynomial \( f(x) = x^2 - i \) factors as \((x - (\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}i))(x + (\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}i))\).

Step 1: Identify the Polynomial and Imaginary Component

The given polynomial function is \( f(x) = x^2 - i \), where \( i \) is the imaginary unit, \( i = \sqrt{-1} \). Our task is to factor this polynomial into a product of two linear complex expressions.

Step 2: Recognize the Structure for Factoring

The polynomial \( x^2 - i \) can be compared to the structure \( a^2 - b^2 = (a+b)(a-b) \), which allows us to use its difference of squares to factor. Here, we need to express \( i \) as a square.

Step 3: Express \( i \) as a Square

We need to write \( i \) in a squared form. Notice that \( i = (\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}i)^2 \). This gives us a way to apply the difference of squares.

Step 4: Factor the Polynomial

Now using the difference of squares form \( x^2 - (i) = x^2 - (\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}i^2) \), we can factor \( f(x) = x^2 - i \) as \((x - (\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}i))(x + (\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}i))\).

Key Concepts

Complex Numbers

Complex numbers are a fascinating area of mathematics, opening a world beyond the real number line. A complex number is the combination of a real part and an imaginary part. The general form of a complex number is \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit. Complex numbers allow us to extend our math toolbox for solving equations that do not have real solutions, like the square root of a negative number.
Key points about complex numbers include:

• They can be plotted on the complex plane, similar to how we plot coordinates on a graph.
• The real part represents the horizontal axis and the imaginary part represents the vertical axis.

In our polynomial exercise, identifying the components and structure involving complex numbers was essential for mastering the factorization. Understanding this broader concept helps in seeing polynomials as expressions that can be broken down even in the presence of complex elements.

Imaginary Unit

The imaginary unit, denoted as \( i \), is a mathematical concept used to extend the real numbers. It is defined by the property \( i^2 = -1 \). This definition allows us to deal with roots of negative numbers, which are not possible within the scope of real numbers.
In our example exercise, \( i \) was used in the given polynomial function \( f(x) = x^2 - i \), representing a simple yet powerful tool in algebra. Here is what makes \( i \) important:

• It helps solve equations that have no real solution, enriching the set of possible solutions.
• Combining \( i \) with real numbers leads to complex numbers, expanding the real number line into a complex plane.
• Operations involving \( i \) follow classic algebraic rules, albeit adjusted for the defining condition of \( i^2 = -1 \).

Knowing how to work with \( i \) and incorporate it into polynomials is a fundamental skill in algebra and calculus.

Difference of Squares

The difference of squares is a technique used frequently in algebra to simplify expressions and solve equations. It applies to expressions of the form \( a^2 - b^2 \), which can be factored into \( (a + b)(a - b) \). This is a specific case of the more general formula known as "difference of two squares."
In our polynomial \( x^2 - i \), recognizing the structure reminiscent of a difference of squares enables simplification even when dealing with complex numbers:

• The first step involves expressing the constant component as a square term, as we did with \( i \).
• Then, factor the polynomial using the difference of squares formula \( a^2 - b^2 \).
• This leads to linear factors that reveal the polynomial's roots or solutions, even in the complex plane.

Understanding and applying the difference of squares is vital, as it provides a strategic approach to breaking down polynomials and gleaning insight into the behavior of more complex functions.