Problem 119 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the expression is a polynomial. If it is, write the polynomial in standard form. $$4 x^{2}+x^{-1}-3$$

Step-by-Step Solution

Answer

No, the expression \(4 x^{2}+x^{-1}-3\) is not a polynomial because not all exponents of the variable \(x\) are nonnegative integers.

Step 1: Identify Variable Exponents

In the provided expression \(4 x^{2}+x^{-1}-3\), identify the exponents of variable \(x\). The given equation has the exponents \(2\) and \(-1\) respectively for \(x\).

Step 2: Verify Nonnegative Integer Exponents

Verify if each of the exponents of variable \(x\) is a nonnegative integer. It can be observed that the exponent \(-1\) for \(x\) in the given expression is not a nonnegative integer.

Step 3: Determine if Expression Is a Polynomial

Since all exponents of \(x\) are not nonnegative integers, the given expression \(4 x^{2}+x^{-1}-3\) is not a polynomial.

Key Concepts

Understanding Standard Form

In mathematics, polynomials are expressions that consist of variables and coefficients, properly arranged. One way to organize polynomials is by placing them in "standard form." In this arrangement, the terms of the polynomial are ordered from the term with the highest degree (the largest exponent) to the lowest degree. For example, a polynomial in standard form might look like: \[ a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \] where \(a_n\), \(a_{n-1}\), ..., \(a_0\) are coefficients and \(n\) is a whole number. This order makes it easier to identify the leading term, which is crucial when performing operations such as addition, subtraction, or division of polynomials. Always remember to arrange polynomial terms in descending order of their exponents when creating the standard form.

Recognizing Nonnegative Integer Exponents

One of the defining characteristics of a polynomial is that all exponents of the variable(s) in the expression must be nonnegative integers. This means we are looking at whole numbers including zero but not negative numbers or fractions. To simplify:

• Nonnegative integers include \(0, 1, 2, ...\).
• An expression like \(x^{-1} \) or \(x^{1/2} \) is not considered a polynomial because their exponents are negative or fractional.

This rule is essential for a polynomial's definition because negative or fractional exponents introduce functions that aren't suitable for polynomial operations and analysis. Checking that all exponents meet this criterion is a critical step in determining if an expression is a polynomial.

Expression Verification

Checking if an expression is a polynomial involves analyzing each term's structure and ensuring compliance with polynomial properties. To conduct an "expression verification," start by spotting all variable terms and their respective exponents:

• Check if the exponents are nonnegative integers.
• Ensure the expression only consists of sums, differences, and multiples of these terms.

For the expression \(4x^2 + x^{-1} - 3\), we see two exponents: 2 (which is acceptable) and \(-1\) (which is not). Because one exponent fails to be a nonnegative integer, this expression does not match the criteria for a polynomial. Thus, thorough verification ensures the proper classification of expressions, making further computation and analysis more manageable.