Problem 3 - Algebra 2

Question

Determine whether each expression is a polynomial. If is a polynomial, state the degree of the polynomial. $$ \frac{m w^{2}-3}{n z^{3}+1} $$

Step-by-Step Solution

Answer

The expression is not a polynomial because it includes a variable in the denominator.

Step 1: Define a Polynomial

A polynomial is an expression formed by adding or subtracting the terms made up of variables raised to non-negative integer exponents and coefficients. It looks like this: \(a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\).

Step 2: Analyze the Given Expression

The given expression is \( \frac{m w^{2}-3}{n z^{3}+1} \). It is written as a fraction with a polynomial in both the numerator \(m w^{2}-3\) and the denominator \(n z^{3}+1\).

Step 3: Check for Polynomial Characteristics

To be a polynomial, an expression must not contain variables in exponents or denominators. Because the expression has a variable in the denominator, specifically \(n z^{3}+1\), it does not meet the polynomial criteria.

Step 4: Conclude the Nature of the Expression

Since dividing by a polynomial disqualifies it from being considered a single polynomial expression, the overall expression \( \frac{m w^{2}-3}{n z^{3}+1} \) is not a polynomial.

Key Concepts

Degree of a Polynomial

The degree of a polynomial is an important concept that helps us understand its characteristics. A polynomial is generally expressed in the form of sums or differences of terms consisting of variables raised to non-negative integer exponents. For example, the polynomial \[a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\] is an expression where each term consists of:

• a coefficient, like \(a_n\), \(a_{n-1}\), etc.,
• one or more variables raised to a whole number power.

The degree of the polynomial is determined by the term with the highest power of the variable. If you have the expression \(5x^3 + 4x^2 - x + 7\), the degree is 3, because \(x^3\) is the term with the highest exponent.
Knowing the degree of a polynomial tells you how many solutions or zero crossings it might have and gives you hints about the graph's shape, such as whether it is a simple curve or a more complex wave. Each degree level provides additional complexity or bends in the graph. Remember, the degree is only considered in the highest terms without any division by another variable.

Characteristics of Polynomials

Understanding the characteristics of polynomials helps distinguish them from other algebraic expressions. Polynomials are characterized by their terms, which include:

• Coefficients
• Variables with whole number exponents
• A sum or difference of these terms

They must not involve fractional or negative exponents and cannot include variables in the denominators or under any radical sign. This allows polynomials to have smooth, continuous curves when graphed, without any breaks or asymptotic behaviors.
Polynomials are also classified by their number of terms:

• A monomial has one term, like \(3x^2\)
• A binomial consists of two terms, such as \(x^3 - 2x\)
• A trinomial includes three terms, like \(2x^2 + x - 5\)

Each type contributes differently to the polynomial's behavior and graph.

Polynomials with Variables in Denominators

When examining whether an expression qualifies as a polynomial, the placement of variables plays a crucial role. A fundamental rule of polynomials is that variables should not appear in denominators. This excludes any fraction where the variable is located in the denominator, such as in rational expressions.
The expression \(\frac{m w^{2}-3}{n z^{3}+1}\) is a prime example of this invalid scenario for polynomials. Despite having polynomials individually in the numerator \(mw^2 - 3\) and the denominator \(nz^3 + 1\), the overall expression is not considered a polynomial.
The reason for this restriction centers on continuity and smoothness—an attribute of polynomial functions when graphed. Allowing variables in denominators would introduce breaks in the continuity of their graphs by creating asymptotes, thus violating one of the core polynomial properties. This distinction is vital in separating polynomial equations from rational functions, where such divisions are allowed and common.