Problem 2 - Algebra and Trigonometry

Question

In Exercises 1–4, is the algebraic expression a polynomial? If it is, write the polynomial in standard form. $$ 2 x+3 x^{-1}-5 $$

Step-by-Step Solution

Answer

No, the algebraic expression \(2x + 3x^{-1} -5 \) is not a polynomial because it contains a term with a negative exponent.

Step 1: Understand What a Polynomial is

A polynomial is an algebraic expression consisting of variables and coefficients. The variables have non-negative integer exponents. It is composed of terms where each term contains a variable raised to a non-negative integer exponent.

Step 2: Analyze the Given Expression

The given expression is \(2x + 3x^{-1} -5 \). It includes a term \(3x^{-1}\) which has a negative exponent.

Step 3: Conclusion

Since the given algebraic expression has a term with a negative exponent, it cannot be categorized as a polynomial. Polynomials can only have terms with non-negative exponents.

Key Concepts

Algebraic Expression

An algebraic expression is a combination of numbers, variables, and mathematical operations like addition, subtraction, multiplication, and division. For example, expressions like \( 3x + 4 \) or \( 5y - 7 + 2z^2 \) are algebraic expressions. These expressions can be as simple or complex as needed.

The main components of an algebraic expression are terms, each term containing numbers and variables. The coefficients are the numerical parts of the terms, and the powers to which the variables are raised are called exponents. It's crucial to understand these components well as they help in identifying other structures, like polynomials, that might be part of the larger expression.

Standard Form

Let's talk about the standard form of polynomials. Writing a polynomial in standard form means ordering the terms of a polynomial from the highest degree to the lowest.

If you look at a polynomial like \( 4x^3 + 2x^2 - 5 \), it is already in standard form because the exponent on \( x \) decreases. This helps in systematically understanding the expression and makes further operations, like addition or subtraction of polynomials, straightforward. Just remember, the degree of the polynomial is the highest exponent of the variable. It's easy; just arrange and you're done!

Negative Exponent

Negative exponents might seem a bit intimidating at first, but they're quite simple. An expression like \( x^{-1} \) is equivalent to \( \frac{1}{x} \). Essentially, a negative exponent indicates that the base is on the reciprocal side of a fraction.

For instance, consider the expression \( 3x^{-1} = \frac{3}{x} \). This transformation shows that \( x \) is in the denominator. When working with polynomials, remember that terms cannot have variables with negative exponents. This differentiates a polynomial from other algebraic expressions with negative exponents.

Non-negative Integer Exponents

Non-negative integer exponents mean the variables in an expression are raised to exponents that are whole numbers like 0, 1, 2, 3, and so on. These exponents define the polynomial's structure and play a key role in determining if an expression is a polynomial.

Consider a term like \( x^2 \). Here, 2 is a non-negative integer exponent. It's as straightforward as remembering to avoid negative numbers for exponents in polynomials. This feature ensures that the terms simplify easily and remain whole, leading to expressions that are comprehensive and easy to manipulate.