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 mathematical fields. Understanding what constitutes a polynomial, and equally importantly, what doesn't, is crucial for success in mathematics and related disciplines. This comprehensive guide will delve into the definition of a polynomial, explore various functions, and definitively identify those that fail to meet the criteria of a polynomial.

Defining a Polynomial: The Essential Characteristics

Before we can identify non-polynomials, we must firmly grasp the definition of a polynomial. A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The key characteristics are:

• Exponents must be non-negative integers: This is arguably the most important characteristic. The exponents of the variables must be whole numbers (0, 1, 2, 3, ...). Fractional or negative exponents immediately disqualify an expression from being a polynomial.

• Variables must have only non-negative integer powers: This reinforces the previous point. Terms like x⁻², x¹/², or √x are not permitted in a polynomial.

• Coefficients can be real or complex numbers: The numbers multiplying the variables (coefficients) can be any real number (like 2, -5, π) or any complex number (like 3 + 2i).

• Finite number of terms: A polynomial must have a finite number of terms. An infinite series, even if it seemingly fits the other criteria, is not considered a polynomial.

• Variables can appear in multiple terms and in various combinations: A polynomial can have many variables, and those variables can be combined and raised to different powers within different terms (e.g., 3x²y + 2xy² - 5x + 7).

Common Functions that are NOT Polynomials: A Detailed Analysis

Now let's explore several examples of functions that are not polynomials, demonstrating why they fail to satisfy the definition:

1. Functions with Negative Exponents:

Consider the function f(x) = 2x⁻³ + 5x² - 7. This function is not a polynomial because of the term 2x⁻³. The exponent -3 is a negative integer, violating the fundamental rule of polynomials having only non-negative integer exponents.

Example: g(x) = 1/x² = x⁻². This is equivalent to x raised to the power of -2, making it non-polynomial.

2. Functions with Fractional Exponents (Roots):

Functions with fractional exponents represent roots. For example, consider h(x) = √x + 4x - 1. This is not a polynomial because √x can be rewritten as x¹/², which has a fractional exponent (1/2). Fractional exponents are not allowed in polynomials.

Example: j(x) = x^(2/3) + 2. The exponent 2/3 is a fraction, making this function non-polynomial.

Example: k(x) = ³√x² = x^(2/3). Similar to the previous example, the fractional exponent disqualifies it.

3. Functions with Variables in the Denominator:

Any function with a variable in the denominator will not be a polynomial. Consider the function:

i(x) = (3x² + 2) / (x - 1). This is a rational function, not a polynomial. The presence of x in the denominator prevents it from being a polynomial.

Example: l(x) = 5 / (x³ + 1). This function has a variable in the denominator, and therefore is not a polynomial.

4. Functions with Infinite Series (or Infinite Number of Terms):

Polynomials, by definition, have a finite number of terms. Functions represented by infinite series, such as Taylor series or Maclaurin series, are not polynomials. These series are used to approximate functions but are not polynomials themselves.

Example: The exponential function, eˣ, is often represented by the infinite series:

1 + x + x²/2! + x³/3! + x⁴/4! + ...

While this series converges to eˣ, it is not a polynomial because it has an infinite number of terms.

5. Functions with Transcendental Functions:

Transcendental functions are functions that are not algebraic; they cannot be expressed as a finite combination of algebraic operations (addition, subtraction, multiplication, division, and raising to an integer power) on variables and constants. These include functions like trigonometric functions, logarithmic functions, and exponential functions (as discussed above).

Example: m(x) = sin(x) + x². While x² is a polynomial, the inclusion of sin(x), a transcendental function, makes the entire function non-polynomial.

Example: n(x) = ln(x) – 5. The natural logarithm, ln(x), is a transcendental function, making this a non-polynomial function.

Example: o(x) = eˣ + 2x. The exponential function eˣ is a transcendental function. Therefore, this combined function is not a polynomial.

Identifying Non-Polynomials: A Step-by-Step Approach

To effectively determine whether a given function is a polynomial, follow these steps:

1. Check the exponents of the variables: Are all the exponents non-negative integers? If any exponent is negative, fractional, or involves a variable (such as xˣ), the function is not a polynomial.

2. Examine the presence of variables in the denominator: Are there any variables in the denominator of a fraction? If yes, it's not a polynomial.

3. Assess the number of terms: Is the number of terms finite? An infinite series is not a polynomial.

4. Look for transcendental functions: Are there any trigonometric functions (sin, cos, tan), logarithmic functions (log, ln), or exponential functions (eˣ) present? If so, it is not a polynomial.

5. Analyze the overall structure: Does the function conform to the basic form of a polynomial: a sum of terms, each term consisting of a coefficient and a variable raised to a non-negative integer power? If not, it is not a polynomial.

Conclusion: Mastering Polynomial Identification

Understanding what defines a polynomial and recognizing common non-polynomial functions are crucial for success in algebra and related mathematical areas. By carefully examining exponents, variable placement, the number of terms, and the presence of any transcendental functions, you can effectively differentiate between polynomials and other mathematical expressions. This knowledge is a cornerstone for further exploration of more advanced mathematical concepts. Remember to always prioritize checking for negative or fractional exponents and variables in denominators – these are the most frequent reasons why a function is not a polynomial.