Problem 96 - Brief Applied Calculus

Question

Which of the following is not a polynomial, and why? $$ x^{2}+\sqrt{2} \quad x^{\sqrt{2}}+1 \quad \sqrt{2} x^{2}+1 $$

Step-by-Step Solution

Answer

The expression \( x^{\sqrt{2}} + 1 \) is not a polynomial because it has a non-integer exponent.

Step 1: Identify Polynomials

A polynomial is an expression consisting of variables (also called indeterminates), coefficients, and non-negative integer exponents of variables. Polynomials can be written in the form: \( a_n x^n + a_{n-1} x^{n-1} + ext{...} + a_1 x + a_0 \).

Step 2: Check the First Expression

The first expression is \( x^2 + \sqrt{2} \). Here, \( x^2 \) is a polynomial term, and adding the constant \( \sqrt{2} \) results in a polynomial, since exponents on \( x \) remain non-negative integers.

Step 3: Analyze the Second Expression

The second expression is \( x^{\sqrt{2}} + 1 \). In this expression, the term \( x^{\sqrt{2}} \) is problematic because the exponent \( \sqrt{2} \) is not an integer. This disqualifies it from being a polynomial.

Step 4: Evaluate the Third Expression

The third expression is \( \sqrt{2} x^2 + 1 \). Here, \( \sqrt{2} \) is simply a coefficient to the term \( x^2 \), which has a non-negative integer exponent. This is a valid polynomial.

Key Concepts

Non-negative Integer Exponents

When talking about polynomials, one of the key features to look out for are the exponents. In polynomials, exponents must be non-negative integers. This means they should be whole numbers like 0, 1, 2, and so on—not fractions or decimals.
Non-negative integer exponents help determine the degree of a polynomial, which is the highest exponent in the expression. For instance, in the polynomial \(3x^5 + 2x^3 + x\), the degree is 5 because 5 is the highest exponent.
This characteristic helps ensure the function is smooth and continuous, without sudden breaks or non-integral fluctuations. On the other hand, expressions with non-integer exponents, such as \(x^{\sqrt{2}}\), are not polynomials since \(\sqrt{2}\) is not an integer, making the function's behavior less predictable.

Coefficients

In any polynomial, coefficients are the numbers that multiply the variables. They play an essential role in defining the polynomial's weight on each term. Each term in a polynomial has a coefficient that can be a positive number, a negative number, zero, or even an irrational number.

• For example, in the term \(5x^3\), 5 is the coefficient.
• In \(\sqrt{2}x^2\), \(\sqrt{2}\) is the coefficient.

Coefficients can tell us important information about the polynomial. A zero coefficient means that the term vanishes. Negative coefficients impact the graph by flipping its orientation across the axis. Meanwhile, irrational coefficients, like \(\sqrt{2}\) or \(\pi\), are perfectly valid and indicate real-world scaling factors, though they appear less frequently in simpler mathematical contexts.

Variables in Polynomials

Variables are the letters in polynomial expressions, and they represent unknown or changeable values. In polynomials, variables are paired with constants to give us terms like \(2x^3\) or \(x^2\). The beauty of variables is their flexibility; they allow polynomials to represent a wide range of situations from simple motions to complex scientific models.
Each polynomial term, like \(a_n x^n\), involves variables raised to a power and multiplied by coefficients. This makes the expressions powerful tools for calculations and projections.
What distinguishes a polynomial from other algebraic expressions is how variables are used: always with non-negative integer exponents. Changing a variable's exponent to a fraction, for example, transforms the expression into something completely different—often shifting it into the realm of continuous, but non-polynomial, functions. Variables give structure to the polynomial and interact with coefficients and powers to describe functions.