Problem 357 - Precalculus

Question

Use a calculator with CAS to answer the questions. Consider \(\frac{x^{4}-k^{4}}{x-k}\) for \(k=1, 2, 3,\) What do you expect the result to be if \(k=4 ?\)

Step-by-Step Solution

Answer

For \( k=4 \), the result is \( x^3 + 4x^2 + 16x + 64 \).

Step 1: Simplify the Expression

The expression given is \( \frac{x^4 - k^4}{x-k} \). Notice this is a difference of powers which can be simplified using the formula for the difference of powers:\[x^4 - k^4 = (x-k)(x^3 + x^2k + xk^2 + k^3)\]This formula directly shows that the expression \( \frac{x^4 - k^4}{x-k} \) simplifies to \( x^3 + x^2k + xk^2 + k^3 \).

Step 2: Substitute Values for k

We know that \( k = 1, 2, 3 \), so we will consider these cases separately and substitute them into the simplified expression:1. For \( k=1 \):\[x^3 + x^2(1) + x(1)^2 + 1^3 = x^3 + x^2 + x + 1\]2. For \( k=2 \):\[x^3 + x^2(2) + x(2^2) + 2^3 = x^3 + 2x^2 + 4x + 8\]3. For \( k=3 \):\[x^3 + x^2(3) + x(3^2) + 3^3 = x^3 + 3x^2 + 9x + 27\]

Step 3: Identify the Pattern

Each of the results from Step 2 is a polynomial of degree 3 with terms increasing in powers of \( k \). This indicates a general polynomial form:\[x^3 + x^2k + xk^2 + k^3\]As observed, the terms are consistent with the general form \( x^3 + ax^2 + bx + c \) where \( a = k, b = k^2, \) and \( c = k^3 \).

Step 4: Predict the Result for k=4

From the observed pattern and the general form, for \( k = 4 \), the expression should become:\[x^3 + x^2(4) + x(4^2) + 4^3 = x^3 + 4x^2 + 16x + 64\]This is based on applying the same substitution logic used for previous values of \( k \).

Key Concepts

Difference of Powers

When dealing with expressions like \(x^4 - k^4\), we encounter a concept known as the **difference of powers**. This concept relates to the idea that we can express the difference between two power terms—such as \(x^4\) and \(k^4\)—in a factored form.
A powerful tool in dealing with these expressions is the formula:

• \(x^n - k^n = (x-k)(x^{n-1} + x^{n-2}k + \dots + xk^{n-2} + k^{n-1})\)

This formula helps to simplify calculative efforts, especially when the degree of the polynomial is large. In our specific exercise, using \(x^4 - k^4\), the formula becomes: \( (x-k)(x^3 + x^2k + xk^2 + k^3)\).
This form allows us to divide the expression by \(x-k\) easily without remaining complexities. Knowing this difference of powers identity reduces calculations and gives insights into polynomial behavior. It shows the significant role algebraic identities play in simplifying complex problems.

Polynomial Division

The operation of dividing one polynomial by another is known as *polynomial division*. Much like long division with numbers, polynomial division lets us express a polynomial in a simpler form, while revealing relationships between polynomials.
Given the expression \(\frac{x^4 - k^4}{x-k}\), it's key to first break down the numerator using the **difference of powers** formula. This simplifies drastically since the term \(x-k\) effectively cancels out with parts of the numerator since it is a common factor.
For \(x^4 - k^4\), dividing by \(x-k\) relies on the formula: \((x-k)(x^3 + x^2k + xk^2 + k^3)\). As a result, the division simplifies directly to the polynomial \(x^3 + x^2k + xk^2 + k^3\). The always-tidy result highlights the power and utility of properly identifying factorization cases—a crucial skill in handling complex algebra problems.

Substitution Method

The **substitution method** is a straightforward yet powerful tool in algebra. It allows us to evaluate polynomials at specific values, revealing patterns and predictable results. For the expression \(x^3 + x^2k + xk^2 + k^3\), substituting specific values for \(k\) helps illustrate how the expression transforms and behaves.
For instance:

• When \(k=1\), the polynomial simplifies to \(x^3 + x^2 + x + 1\).
• When \(k=2\), it becomes \(x^3 + 2x^2 + 4x + 8\).
• When \(k=3\), it results in \(x^3 + 3x^2 + 9x + 27\).

These steps showcase the gradual increase in each term as \(k\) increases, cementing the value of having substitution as a tool to derive exact algebraic outcomes. By predicting expressions like this at higher values, such as \(k=4\), we successfully anticipate the form \(x^3 + 4x^2 + 16x + 64\), cementing our understanding of the polynomial's scalability and predictability across different scenarios.