Problem 50 - Precalculus with Calculus Previews

Question

If \(n>2\) is an even positive integer, explain why \(f(x)=x^{n}-c^{n}\) can be written as a product of three factors.

Step-by-Step Solution

Answer

Factored as \((x-c)(x+c)(x^{n-2} + x^{n-3}c + ... + c^{n-2})\).

Step 1: Use difference of squares

Start with the expression \(f(x) = x^n - c^n\). Since we know \(n\) is even, we can write \(n = 2m\) where \(m\) is a positive integer. The expression becomes \(x^{2m} - c^{2m}\).This can be written as a difference of squares: \((x^m)^2 - (c^m)^2\).Thus, it can be factored as: \[ (x^m - c^m)(x^m + c^m) \]

Step 2: Factor further using difference of cubes

Since \(m > 1\), continuously apply the difference of squares or sums if possible. Next, apply the difference of cubes formula: For \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\).When examining \((x^m - c^m)\) and \((x^m + c^m)\), check each factor for forms which can still be factored. Potentially factor further into: - \((x-c)(x^2 + xc + c^2)\)- \((x+c)(x^2 - xc + c^2)\) This step ensures deeper breakdown using any recognized algebraic identities.

Key Concepts

Difference of Squares

The concept of the "difference of squares" is fundamental in algebra. It is used to simplify expressions that take the form of two squared terms being subtracted from one another.
The formula for this is:

• \[ a^2 - b^2 = (a - b)(a + b) \]

You identify the difference of squares when you have an expression like \( x^{2m} - c^{2m} \). Here, both terms \( x^m \) and \( c^m \) are squared.
Given the rule, you can rewrite it as \( (x^m - c^m)(x^m + c^m) \). This creates a new expression with two factors.
Recognizing a difference of squares helps in breaking down complex expressions, making them easier to handle.

Difference of Cubes

The "difference of cubes" is another helpful algebraic identity. It allows you to factor expressions where one cube is subtracted from another.
The formula is:

• \[ a^3 - b^3 = (a-b)(a^2 + ab + b^2) \]

When dealing with the problem \( f(x) = x^n - c^n \), once you have expressed it as a product of two factors using difference of squares, you can check further. For example, if within \( (x^m - c^m) \), these terms themselves can be broken down using difference of cubes.
Applying such identities helps in further simplifying and factoring complex polynomial expressions.
These identities open doors to deeper breakdowns of expressions, making intricate polynomial problems more approachable.

Even Integers

Even integers are integers that are divisible by 2 without any remainder. In algebra, recognizing even numbers is useful for specific operations and simplifications.
When an even integer \( n \) is expressed, it can be written in the form \( n = 2m \) where \( m \) is an integer.
This property is significant in problems involving powers, as seen in \( x^{n} - c^{n} \) where \( n = 2m \). By expressing it using "difference of squares," the role of even integers simplifies the expression into manageable factors.
Understanding even numbers helps avoid mistakes during factorization and opens paths for utilizing other algebraic identities effectively.

Algebraic Identities

Algebraic identities are preset formulas or equations deemed always true for all variables involved. They are powerful tools in simplifying algebraic expressions.
They include, but aren't limited to:

• The difference of squares: \( a^2 - b^2 = (a-b)(a+b) \)
• The difference of cubes: \( a^3 - b^3 = (a-b)(a^2 + ab + b^2) \)

Using these identities transforms complex polynomials into simpler, factored components, which are easier to handle and solve.
Recognizing when and how to apply these identities in an expression, such as \( f(x) = x^n - c^n \), is crucial for effective problem-solving in algebra.
They serve as shortcuts and reveal insights into the nature of polynomials and their behavior, providing a structured pathway to solution strategies.