Problem 19 - Algebra 2

Question

Determine whether each expression is a polynomial. If it is a polynomial, state the degree of the polynomial. $$ 5 x^{2} y^{4}+x \sqrt{3} $$

Step-by-Step Solution

Answer

The expression is not a polynomial due to the term \(x\sqrt{3}\).

Step 1: Identify Terms in the Expression

The given expression is \(5x^2y^4 + x\sqrt{3}\). First, let's identify each term: \(5x^2y^4\) and \(x\sqrt{3}\).

Step 2: Analyze the First Term

The term \(5x^2y^4\) has variables \(x\) and \(y\) with respective powers of 2 and 4. This term is a valid polynomial term.

Step 3: Analyze the Second Term

The term \(x\sqrt{3}\) can be rewritten as \(x \cdot 3^{1/2}\). Since there is a variable multiplied by a non-integer exponent when expressed like this, it is not a valid polynomial term.

Step 4: Determine If the Expression Is a Polynomial

Since \(x\sqrt{3}\) is not a valid polynomial term due to the non-integer exponent, the entire expression \(5x^2y^4 + x\sqrt{3}\) is not a polynomial.

Key Concepts

Polynomial Degree

When discussing polynomials, the degree is an important characteristic. It tells us about the highest power of any term in the polynomial when all similar terms are combined.A single-variable polynomial's degree is determined by the term with the highest exponent. However, things can get a little more complicated when dealing with multivariable polynomials.For example, if you have the polynomial expression \(5x^2y^4\), the degree is calculated by adding the powers of each variable within a term:

• \(x^2\) contributes 2 to the degree.
• \(y^4\) contributes 4 to the degree.

So, the degree of the term \(5x^2y^4\) is \(2 + 4 = 6\).The degree of a multivariable polynomial is determined by the term with the highest total degree (sum of the exponents within that term). It can help identify the number of potential roots or the behavior of the polynomial graph.

Multivariable Polynomials

Multivariable polynomials involve more than one variable, like \(x\) and \(y\) in the expression \(5x^2y^4\). These polynomials are a sum of terms, and each term is a product of variables raised to non-negative integer powers.In multivariable polynomials, each term is examined in isolation to identify its degree, as well as the polynomial's overall degree:- Consider the example term \(5x^2y^4\):

• It involves two variables: \(x\) and \(y\).
• Each variable can contribute separately to the degree.
• The combined power for this term is \(6\), calculated by adding each variable's exponent.

Multivariable polynomials might seem complex, but they allow for modeling relationships where two or more variables interact. These relationships are vital in fields such as physics, economics, and engineering.

Non-polynomial Terms

Not all expressions that look like polynomials meet the criteria to be considered as such. Polynomial terms must only include non-negative integer exponents, meaning any fraction or negative exponents disqualify them.Take the expression \(x\sqrt{3}\):- Rewritten as \(x \cdot 3^{1/2}\), it introduces an exponent that is a fraction, \(\frac{1}{2}\), which means it is not a polynomial term.This distinction is crucial because only polynomials possess certain properties that make them easier to graph and analyze mathematically. Non-polynomial terms, due to their fractional or negative exponents, usually indicate more complex and different kinds of functions that behave differently from polynomials.In conclusion, understanding what disqualifies an expression from being a polynomial helps clarify the unique traits that genuine polynomials have in terms of predictability and tractability.