Problem 8 - Algebra and Trigonometry

Question

Determine whether the expression is a polynomial. If it is, state its degree. $$\frac{2}{x^{2}-4 x+6}$$

Step-by-Step Solution

Answer

The expression is not a polynomial.

Step 1: Identify the expression type

The given expression is \( \frac{2}{x^{2}-4x+6} \). It is written in the form of a fraction.

Step 2: Analyze the denominator

Examine the denominator \( x^{2}-4x+6 \), which is a quadratic expression, meaning it is a polynomial of degree 2. However, since the expression is in the form of a fraction, the whole expression is not a polynomial.

Step 3: Determine if the expression is a polynomial

A polynomial is a sum of terms containing variables raised to non-negative integer powers, with coefficients that are real numbers. Polynomials do not include fractions with variable expressions in the denominator. Since \( \frac{2}{x^{2}-4x+6} \) has a variable expression in the denominator, it is not a polynomial.

Key Concepts

Expression Analysis

When trying to determine if an expression is a polynomial, one of the first steps is to analyze what type of expression you are dealing with. Let's break it down:

• If the expression is written as a fraction, like this problem, pay close attention to both the numerator and the denominator. The presence of variables in these parts is crucial for identifying the essence of the expression.
• In the exercise, the expression given is \( \frac{2}{x^2 - 4x + 6} \). The form of a fraction signals that special consideration is needed regarding polynomials.
• Each part of the fraction must be checked independently. This means reviewing both the one in the numerator and the one in the denominator to decide if either or both are polynomials themselves.

By understanding these initial checks, we can progress to more detailed insights into identifying polynomials.

Degree of Polynomial

The degree of a polynomial is a key concept in algebra and can be easily identified once you know what to look for.

• The degree of a polynomial is the greatest power of the variable in the expression. For example, in the expression \( x^2 - 4x + 6 \), the term with the highest exponent is \( x^2 \), making it a quadratic polynomial of degree 2.
• To accurately identify the degree, ensure the polynomial is written in its simplest form with terms arranged in descending order of powers.
• For any polynomial, remember that all coefficients should be real numbers, and each power should be a non-negative integer.

These guidelines are blocked out easily by analyzing the highest powered term in your expression, confirming its degree swiftly.

Non-Polynomial Expression

Understanding what makes an expression a non-polynomial is equally important in mathematics.

• A key characteristic of non-polynomial expressions is the presence of variables in denominators, like in the expression \( \frac{2}{x^2 - 4x + 6} \).
• While the numerator could be a polynomial, if the entire expression includes a division by a variable-containing expression, it disqualifies as a polynomial.
• Additionally, polynomials do not have negative exponents or variables inside functions like square roots or fractional powers.

In conclusion, any expression that includes these features or forms does not meet the criteria for polynomials, being categorized as non-polynomial expressions instead.