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Chapter 5: Problem 98

What is a polynomial in two variables? Provide an example with your description.

Short Answer

Expert verified
A polynomial in two variables is an algebraic expression composed of the sum of several terms, where each term consists of the product of a constant and whole powers of two variables. An example would be \(f(x, y) = 2x^{2}y + 3xy^{2} - 7x + 4y + 9\).

Step by step solution

01

Define a Polynomial in Two Variables

A polynomial in two variables is defined as an algebraic expression that involves two different variables, usually taken as \(x\) and \(y\). The terms of this polynomial can be of any degree (a non-negative integer), but the power of each variable in the terms must be whole. That is, fractional or negative powers are not allowed.
02

Structure of a Polynomial in Two Variables

In general, the polynomial in two variables can be expressed as: \(f(x, y) = a_{n}x^{n}y^{m}+ a_{n-1}x^{n-1}y^{m-1}+...+a_{1}x+ a_{0}\), where \(a_{n}\), \(a_{n-1}\), ... , \(a_{1}\), and \(a_{0}\) are real numbers (constants), and \(n\) and \(m\) are non-negative whole numbers.
03

Provide an Example

An example of a polynomial in two variables is \(f(x, y) = 2x^{2}y + 3xy^{2} - 7x + 4y + 9\). This polynomial contains five terms with both \(x\) and \(y\) variables in non-negative integer power. The degree of this polynomial would be determined by the term with the highest sum of powers of both variables, in this case, it's degree 3.

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Key Concepts

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

Algebraic Expressions
Algebraic expressions are mathematical phrases combining numbers, variables, and operational symbols. They don't have an equality sign like an equation does.

In algebraic expressions, variables such as \(x\) and \(y\) are used to represent numbers. They allow us to generalize mathematical relationships and solve problems with unknown values. For example, in the expression \(2x + 3\), \(2\) is a coefficient of the variable \(x\), and \(3\) is a constant.
  • Coefficients are numbers that multiply the variables.
  • Constants are fixed values added to the expressions.
  • Operators like \(+\) and \(-\) show how variables and numbers interact.
Understanding algebraic expressions is essential before moving on to more complex structures like polynomials.
Two Variables
When dealing with polynomials in two variables, we work with expressions involving two different variables, usually denoted \(x\) and \(y\). This dual-variable setup allows us to represent more complex scenarios by finding relationships between two quantities.

In two-variable polynomials, each term combines powers of \(x\) and \(y\). For example, in the polynomial \(2x^2y + 3xy^2\), each term involves both variables.
  • The term \(2x^2y\) has \(x\) raised to the power of 2 and \(y\) to the power of 1.
  • The term \(3xy^2\) has \(x\) to the power of 1 and \(y\) to the power of 2.
Such polynomials are useful in various fields like physics and engineering, where they model relationships involving two different variables.
Polynomial Degree
The degree of a polynomial in two variables is a key concept. It reflects the highest sum of the exponents of the variables in any term of the polynomial.

In the example \(f(x, y) = 2x^2y + 3xy^2 - 7x + 4y + 9\):
  • The term \(2x^2y\) has an exponent sum of \(2 + 1 = 3\).
  • The term \(3xy^2\) also has an exponent sum of \(1 + 2 = 3\).
The degree of this polynomial is 3, determined by these terms with the highest sum of exponents. Understanding the degree helps determine the behavior of the polynomial graphically and its complex relationships when solving equations.
Educational Example
Exploring polynomials with an example can solidify understanding. Consider the polynomial \(f(x, y) = 2x^2y + 3xy^2 - 7x + 4y + 9\).

This expression includes five distinct terms:
  • \(2x^2y\), emphasizing the contribution of both variables \(x\) and \(y\).
  • \(3xy^2\), where both variables again come together, manipulating their powers strategically.
  • Even simple terms like \(-7x\) and \(4y\) show how variables can stand alone.
  • The constant term \(9\) adds quantity without involving variables.
Using examples like this makes the abstract idea of polynomials concrete, preventing confusion about complex terms and structures. Through examples, students can see patterns and enhance their polynomial fluency.

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Most popular questions from this chapter

When dividing a polynomial by a binomial, explain when to stop dividing.

Explain the difference between \((-7)^{0}\) and \(-7^{0}\).

What is \(6 \%\) of \(20 ?\) (Section \(2.4,\) Example 5 )

Explain how to divide monomials. Give an example.

Use a graphing utility to graph each side of the equation in the same viewing rectangle. (Call the left side \(y_{1}\) and the right side \(y_{2} .\) I If the graphs coincide, verify that the multiplication has been performed correctly. If the graphs do not appear to coincide, this indicates that the multiplication is incorrect. In these exercises, correct the right side of the equation. Then graph the left side and the corrected right side to verify that the graphs coincide. \((x+1)(x-1)=x^{2}-1 ;\) Use a \([-6,5,1]\) by \([-2,18,1]\) viewing rectangle.
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