Which Of The Following Are Not Polynomials

Which of the Following are Not Polynomials? A Comprehensive Guide

Polynomials are fundamental building blocks in algebra and numerous other areas of mathematics. Understanding what constitutes a polynomial and, equally importantly, what doesn't, is crucial for success in mathematical studies. This comprehensive guide will explore the definition of a polynomial and provide numerous examples of expressions that are not polynomials, explaining why they fail to meet the criteria.

Defining a Polynomial

Before we delve into examples of non-polynomials, let's firmly establish the definition. A polynomial is an expression consisting of variables (often denoted by x, y, z, etc.) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

Key Characteristics of a Polynomial:

• Non-negative integer exponents: The exponents of the variables must be whole numbers (0, 1, 2, 3, and so on). Fractional or negative exponents are not allowed.
• Variables in the numerator only: Variables cannot appear in the denominator of a fraction.
• No transcendental functions: Functions like sine (sin), cosine (cos), tangent (tan), exponential (e^x), logarithmic (log x), or absolute value (|x|) are not permitted within a polynomial.
• Finite number of terms: A polynomial must have a finite (limited) number of terms.

Common Examples of Non-Polynomials

Now, let's examine several categories of mathematical expressions that are frequently mistaken for polynomials but fail to satisfy the definition:

1. Expressions with Negative Exponents

Any expression containing a variable raised to a negative exponent is not a polynomial.

Examples:

• 3x^-2 + 5x: The term 3x^-2 has a negative exponent (-2), disqualifying the entire expression. Remember, x^-2 is equivalent to 1/x².
• (x + 2)^-1: This is equivalent to 1/(x+2), which contains a variable in the denominator.
• 7/x³ - 4x + 2: The term 7/x³ is the same as 7x^-3, which features a negative exponent.

2. Expressions with Fractional Exponents

Similarly, expressions with variables raised to fractional exponents (other than integers) are not polynomials.

Examples:

• x^1/2 + 2x - 1: The term x^1/2 is equivalent to √x, which is a radical expression, not allowed in polynomials.
• 4x^2/3 - 5: The fractional exponent 2/3 disqualifies this expression from being a polynomial.
• (x² + 1)^1/3: This expression contains a fractional exponent applied to an entire expression.

3. Expressions with Variables in the Denominator

Any expression where a variable appears in the denominator of a fraction is not a polynomial.

Examples:

• 5/x + 2x² - 7: The variable x is in the denominator of the fraction 5/x.
• (x + 1)/(x - 1): Both the numerator and denominator contain variables, rendering this expression non-polynomial.
• 1/(x² + 4): The presence of x in the denominator immediately rules this out as a polynomial.
• (x² + 3x)/(x + 2): The presence of x in the denominator prevents this from being a polynomial

4. Expressions with Variables Under a Radical Sign (Roots)

Expressions with variables under a radical sign (square root, cube root, etc.) are generally not polynomials, unless the radical simplifies to an integer exponent.

Examples:

• √x + 3: The square root of x (x^1/2) is a fractional exponent, violating the polynomial definition.
• ∛(x² + 1): This is equivalent to (x² + 1)^1/3, containing a fractional exponent.
• √(x³): While this simplifies to x^3/2, the fractional exponent means it's not a polynomial. However, expressions like √16 which simplifies to 4, are allowed as this simplifies to a constant, and constants are allowed in polynomials.

5. Expressions Involving Transcendental Functions

Expressions containing trigonometric functions (sine, cosine, tangent, etc.), exponential functions (e^x), logarithmic functions (ln x or log x), or absolute value functions are not polynomials.

Examples:

• sin x + 2x - 1: The sine function (sin x) is a transcendental function.
• e^x + 5: The exponential function e^x is not permitted in polynomials.
• ln(x + 1): The natural logarithm function (ln) is a transcendental function.
• |x| + 2x: The absolute value function |x| is also not part of polynomial expressions.
• 2^x + x²: The exponential term 2^x disqualifies the expression.

6. Expressions with Infinitely Many Terms

Polynomials have a finite number of terms. An expression with infinitely many terms is not a polynomial. While you won't often encounter such an expression explicitly, the concept is important for understanding the limits of the polynomial definition.

Example:

The infinite series 1 + x + x² + x³ + x⁴ + ... is not a polynomial because it has infinitely many terms. This is actually a geometric series.

7. Expressions with Variables in the Exponent

An expression with a variable in the exponent is not a polynomial.

Examples:

• x^x: The variable x is in the exponent.
• 2^x: The base is a constant, but the exponent is a variable.
• x^(x+1): The exponent is itself a polynomial expression containing a variable.

Identifying Non-Polynomials: A Practical Approach

When faced with an expression, use this checklist to determine if it's a polynomial:

1. Check the exponents: Are all exponents of the variables non-negative integers? If not, it's not a polynomial.
2. Check the denominator: Are there any variables in the denominator of a fraction? If yes, it's not a polynomial.
3. Check for transcendental functions: Are there any trigonometric, exponential, logarithmic, or absolute value functions present? If yes, it's not a polynomial.
4. Count the terms: Is the number of terms finite? If not, it's not a polynomial.
5. Check for variables in exponents: Are there any variables in the exponents? If yes, it is not a polynomial.

By systematically applying these checks, you can confidently identify whether a given mathematical expression qualifies as a polynomial or falls into the category of non-polynomial expressions. This understanding is crucial for various mathematical procedures and problem-solving techniques. Remember, polynomials possess a specific structure and limitations, and deviating from those rules automatically excludes an expression from the polynomial family.