Which Is True About The Polynomial 3xy2 5x2y

Which is true aboutthe polynomial 3xy² + 5x²y?

The expression 3xy² + 5x²y is a compact representation of a two‑term polynomial that appears frequently in algebra, physics, and economics. Although it looks simple, the polynomial hides several structural properties that determine its behavior under various operations. This article unpacks those properties step by step, clarifies common misconceptions, and answers the central question: **which statements about the polynomial are actually true?

Understanding the Basic Structure

The polynomial consists of two monomials:

• 3xy² – a term where the coefficient is 3, the variable x appears to the first power, and y appears squared.
• 5x²y – a term where the coefficient is 5, x is squared, and y appears to the first power.

Both terms share the same total degree (three), but they differ in the distribution of exponents between the variables. Recognizing this distinction is essential when evaluating claims about the polynomial’s symmetry, factorability, or degree.

Simplifying and Rearranging Terms

Although the expression is already in its simplest expanded form, it can be rearranged to highlight common factors: - Factor out the greatest common factor (GCF).
The GCF of the two monomials is xy. Pulling xy out yields:

[ xy,(3y + 5x) ]

• Reorder the terms by descending powers of x.
  Writing the polynomial as 5x²y + 3xy² emphasizes the higher power of x first, which is the conventional order in many textbooks.

These manipulations do not change the value of the expression; they only provide different perspectives that can be useful for further algebraic work.

Factoring the Expression

Factoring reveals hidden relationships between the terms. Starting from the GCF extraction above, we obtain: [ \boxed{xy,(3y + 5x)} ]

This factored form is irreducible over the integers because the binomial (3y + 5x) cannot be broken down further without introducing fractions. Still, the factorization is valuable for:

• Solving equations. Setting the polynomial equal to zero gives xy (3y + 5x) = 0, leading to the solution set x = 0, y = 0, or 3y + 5x = 0.
• Graphical analysis. The zero‑sets correspond to the coordinate axes and a straight line, offering a clear visual interpretation.

Determining Degree and Number of Terms

• Total degree: Each monomial has exponents that sum to 3 (1 + 2 = 3 and 2 + 1 = 3). So, the polynomial is a cubic (degree‑3) expression.
• Number of non‑zero terms: There are exactly two non‑zero terms, making it a binomial.
• Number of distinct variables: Two variables (x and y) appear, so the polynomial is bivariate.

These characteristics answer many “which is true” queries: the polynomial is cubic, binomial, and bivariate.

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Which Statements Are Actually True?

Below is a concise checklist that addresses typical true/false questions about the polynomial. Each item is accompanied by a brief justification.

1. The polynomial can be written as xy(3y + 5x).
   True. This is the result of factoring out the GCF xy.

2. The polynomial has a degree of 2.
   False. The highest combined exponent of any term is 3, so the degree is 3.

3. Both terms contain the same power of x.
   False. The first term has x¹, while the second term has x².

4. The polynomial is symmetric with respect to swapping x and y.
   False. Swapping x and y transforms the expression into 3yx² + 5y²x, which is not equivalent to the original.

5. The polynomial can be simplified to a single monomial.
   False. Because the two monomials are not like terms, they cannot be combined into one term.

6. The polynomial’s graph passes through the origin (0, 0).
   True. Substituting x = 0 or y = 0 makes the entire expression zero, so the origin is a root.

7. The polynomial is divisible by x + y.
   False. Performing polynomial division shows a non‑zero remainder, indicating that x + y is not a factor.

8. The coefficients 3 and 5 are relatively prime.
   True. The greatest common divisor of 3 and 5 is 1, meaning they share no common factors other than 1.

These statements illustrate how careful inspection of exponents, factors, and symmetry determines the validity of claims about the polynomial.

Common Misconceptions

Even though the polynomial is straightforward, several misconceptions frequently arise: - Misconception: “Because the polynomial has two terms, it must be a quadratic.”
Reality: The number of terms does not dictate the degree; the exponents do. Here, despite being a binomial, the degree is cubic.

• Misconception: “The polynomial is symmetric, so swapping x and y leaves it unchanged.”
  Reality: Symmetry would require the expression to remain identical after swapping, which is not the case for 3xy² + 5x²y.

• Misconception: “Factoring always reduces the number of terms dramatically.”
  Reality: In this example, factoring merely extracts a common factor,