Problem 3 Determine whether each expressio... [FREE SOLUTION]

Chapter 13: Problem 3

Determine whether each expression is a polynomial. Explain your reasoning. If it is, classify it as a monomial, binomial, or trinomial. $$x^{2}+x y^{2}-y^{2}$$

Short Answer

Expert verified

The expression $ x^2 + xy^2 - y^2 $ is a polynomial and is classified as a trinomial.

Step by step solution

Define a Polynomial

A polynomial is an expression made up of variables, coefficients, and exponents. Each term is a product of a constant and a variable raised to a non-negative integer power. Polynomials cannot have variables in the denominator, negative exponents, or variables under root signs.

Analyze the Given Expression

The given expression is $ x^2 + xy^2 - y^2 $. It consists of three terms: $ x^2 $, $ xy^2 $, and $ -y^2 $. These have constant coefficients of 1, 1, and -1 respectively, and each variable is raised to a non-negative integer power.

Evaluate Each Term

Look at each term individually: $ x^2 $ has two as the exponent on $ x $, $ xy^2 $ has one on $ x $ and two on $ y $, and $ -y^2 $ has two on $ y $. All the exponents are non-negative integers, which fits the criteria of a polynomial.

Classify the Polynomial

Since the expression has three terms, it is classified as a trinomial. Each term meets the criteria for a polynomial, confirming the expression is indeed a polynomial.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Expression Classification

Understanding how to classify a mathematical expression is a fundamental skill in algebra. Expressions are combinations of numbers, variables, and operators (like addition or subtraction signs). When classifying, we primarily look at the terms. Each term is a piece of the expression separated by `+` or `-`. For polynomials, the classification changes with the number of terms. An expression with one term is a **monomial**, while two terms make it a **binomial**. If an expression has three distinct terms, it is a **trinomial**. Classifying expressions helps in determining how to simplify or apply operations to them.

Trinomial

A trinomial is specifically a type of polynomial with exactly three terms. For example, the expression $ x^2 + xy^2 - y^2 $ is classified as a trinomial. Each term must be a product of a constant and variables raised to non-negative integer powers to ensure it's part of a polynomial.

The term $ x^2 $ is made up of the variable $ x $ raised to the power of 2.
The middle term $ xy^2 $ involves both $ x $ and $ y $ where $ y $ is squared.
The last term $ -y^2 $ consists of the variable $ y $ raised to the power of 2, and a coefficient of -1.

Understanding the structure of a trinomial helps in tasks like factoring or solving equations.

Mathematical Expressions

Mathematical expressions are the language of math. They are used to represent numbers, operations, and variables in a concise way. Expressions can be as simple as a single number or variable, or they can be complex with multiple components.

Each mathematical expression consists of terms. Terms are parts of the expression separated by operators (`+` or `-`).
Understanding expressions involves recognizing how they combine algebraic symbols into a coherent piece.
Many expressions, including polynomials, are used to model real-world situations and solve equations.

By breaking down an expression into its terms, you gain a better understanding of its structure and how to manipulate it.

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91影视

Determine whether each expression is a polynomial. Explain your reasoning. If it is, classify it as a monomial, binomial, or trinomial. $$x^{2}+x y^{2}-y^{2}$$

Short Answer

Step by step solution

Define a Polynomial

Analyze the Given Expression

Evaluate Each Term

Classify the Polynomial

Key Concepts

Expression Classification

Trinomial

Mathematical Expressions

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