Problem 10 - Calculus: One and Several Variables

Question

State whether the function is a polynomial. a rational function (but not a polynomial), or neither a polynomial nor a rational function. If the function is a polynomial, give the degree. $$h(x)=\frac{(\sqrt{x}+2)(\sqrt{x}-2)}{x^{2}+4}$$.

Step-by-Step Solution

Answer

The given function \(h(x)=\frac{\sqrt{x}+2)(\sqrt{x}-2)}{x^{2}+4}\) is a rational function but not a polynomial.

Step 1: Simplify the numerator

Firstly, simplify the numerator by multiplying the bracket terms. It forms a difference of squares \(a^{2} - b^{2}\). The numerator simplifies to \(x - 4\) after calculation.

Step 2: Define the expression

The expression now is \(h(x)=\frac{x-4}{x^{2}+4}\). The top part \(x - 4\) is a polynomial of degree 1. The bottom part \(x^{2} + 4\) is a polynomial of degree 2.

Step 3: Identify the nature of the entire expression

Since the entire expression is a fraction of two polynomials, it is a rational function. However, it cannot be simplified to a single polynomial as doing so would involve taking the square root of \(x\), introducing a non-polynomial term, thereby it is not a polynomial.

Key Concepts

Polynomial Degree

The degree of a polynomial is a core concept in understanding its behavior and representation. In mathematics, the degree is determined by the highest power of the variable in the polynomial expression. For example, in the expression \(x^2 + 4\), the degree is 2 because the highest power of \(x\) is 2.
In the exercise given, we have two separate polynomial components:

• The numerator: \(x - 4\), which is a polynomial of degree 1.
• The denominator: \(x^2 + 4\), a polynomial of degree 2.

Understanding the degree helps us predict the graph's end behavior and the possible number of roots. It's crucial to identify the degree of each part if you are working with rational functions.

Difference of Squares

The concept of 'difference of squares' comes into play when simplifying expressions like \((\sqrt{x}+2)(\sqrt{x}-2)\).
This is a special algebraic identity where you have the expression \(a^2 - b^2\) which can be simplified to \((a-b)(a+b)\).

In our exercise:

• \(a = \sqrt{x}\) and \(b = 2\), giving \((\sqrt{x}+2)(\sqrt{x}-2)\).
• Using the identity: \((\sqrt{x})^2 - 4 = x - 4\).

This simplification is key as it reduces complex multiplication into a simpler form, making it easier to handle within rational functions. Remember that knowing such identities can significantly speed up your calculations.

Function Simplification

Function simplification helps in understanding and analyzing more complex functions. After simplifying the numerator in our function, we have the expression \(h(x)=\frac{x-4}{x^2+4}\).
The simplification process involves reducing expressions to their most basic form, if possible.

Here are the steps:

• Simplify the numerator using known identities like the difference of squares.
• Identify the polynomials in both the numerator and denominator.
• Check if the expression can be further simplified.

In this context, the function \(h(x)\) can't be simplified into a single polynomial because the original expression involves \(\sqrt{x}\), which introduces a non-polynomial component. Therefore, while the function is a rational function (a ratio of polynomials), it is not a pure polynomial itself. Understanding simplification helps in recognizing the structure and type of mathematical expressions.