Problem 15 - Algebra Form and Function

Question

Is the expression a polynomial in the given variable? $$ \left(a+\frac{1}{x}\right)^{2}-\left(a-\frac{1}{x}\right)^{2}, \text { in } x $$

Step-by-Step Solution

Answer

Answer: No, the expression is not a polynomial in x.

Step 1: Expand the expression

To make the expression easier to analyze, first expand the expression using the identity \((A - B)^2 - (A + B)^2 = -4AB\): $$ \left(a+\frac{1}{x}\right)^{2}-\left(a-\frac{1}{x}\right)^{2} = -4\left(\left(a+\frac{1}{x}\right)\left(a-\frac{1}{x}\right)\right) $$

Step 2: Simplify the expanded expression

Now, simplify the product inside the parentheses: $$ \left(a+\frac{1}{x}\right)\left(a-\frac{1}{x}\right) = a^2 - \left(\frac{1}{x}\right)^2 = a^2 - \frac{1}{x^2} $$

Step 3: Substitute the simplified expression back

Substitute the simplified expression back into the original expression: $$ -4\left(\left(a+\frac{1}{x}\right)\left(a-\frac{1}{x}\right)\right) = -4\left(a^2 - \frac{1}{x^2}\right) = -4a^2 + \frac{4}{x^2} $$

Step 4: Check if the expression fits the definition of a polynomial

Now that we have our simplified expression, let's check if it fits the definition of a polynomial in the variable x. A polynomial should have the form: $$ P(x) = a_0 + a_1x + a_2x^2 + ... + a_nx^n, $$ where \(a_0, a_1, ..., a_n\) are constants and \(n\) is a non-negative integer. Our expression is: $$ -4a^2 + \frac{4}{x^2} $$ Notice that it contains the term \(\frac{4}{x^2}\), which is not in the form of \(a_ix^i\) for any non-negative integer \(i\). Therefore, the expression is not a polynomial in x.

Key Concepts

Polynomial Expressions

Polynomial expressions are a fundamental concept in algebra. Essentially, a polynomial is made up of terms that are combined by addition or subtraction. Each term includes a variable raised to a non-negative integer power and multiplied by a constant coefficient. Polynomials can have one or more terms, depending on how many pieces are in the expression. For example, the expression \(3x^2 + 2x + 1\) is a polynomial with three terms.

To determine if an expression is a polynomial, we need to look for key features:

• The exponents of the variable must be whole numbers (non-negative integers).
• The coefficients should be real numbers.
• Polynomials cannot have variables in the denominator; the terms should not include negative powers of variables.

When analyzing the expression \(\left(a+\frac{1}{x}\right)^{2}-\left(a-\frac{1}{x}\right)^{2}\), it initially appears as a complex statement. However, through examining its terms, particularly looking at powers of \(x\), we can determine it doesn't fit the criteria of being a polynomial, as shown in the analysis.

Variable Analysis

Variable analysis is the process of examining how variables are used within an expression. This involves looking at the power of each variable and understanding how this influences the nature of the expression.

In the expression \(-4a^2 + \frac{4}{x^2}\), we focus on the variable \(x\). The term \(\frac{4}{x^2}\) includes the variable \(x\) in the denominator, implying a negative exponent when rewritten as \(4x^{-2}\). This is important because the definition of a polynomial demands that all variables have non-negative integer powers.

Thus, through this analysis, we establish that the presence of \(x\) as a denominator invalidates the polynomial classification for this particular expression relative to \(x\). Using this step aids in clearly understanding why some expressions resist fitting neatly into the polynomial category.

Expression Simplification

Expression simplification is a critical step in evaluating mathematical statements. The goal is to reduce complex expressions to simpler forms while maintaining their original value.

In our original expression \(\left(a+\frac{1}{x}\right)^{2}-\left(a-\frac{1}{x}\right)^{2}\), the simplification begins by applying the identity \((A - B)^2 - (A + B)^2 = -4AB\). This helps us transform and manage the calculation process more effectively. By expanding it through this identity, the complexities of \(\left(a+\frac{1}{x}\right)^2\) and \(\left(a-\frac{1}{x}\right)^2\) are unpacked and simplified.

• This expansion leads to combining like terms and reducing them into simpler manageable forms like \(a^2 - \frac{1}{x^2}\).
• Further reduction gives \(-4(a^2 - \frac{1}{x^2})\), which simplifies to \(-4a^2 + \frac{4}{x^2}\).

Such simplifications are powerful tools in determining properties of expressions such as whether they conform to polynomial rules. In this case, simplification reveals key terms that don't align with polynomial criteria due to the presence of a denominator variable.