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Chapter 0: Problem 102

Explain how to find the degree of a polynomial in two variables.

Short Answer

Expert verified
The degree of a polynomial in two variables is found by identifying the term with the highest sum of exponents of the variables. In the example \(3x^2y + x^3 + 4xy^2\), all terms have a degree of 3. Therefore, the degree of this polynomial is 3.

Step by step solution

01

Understand the Definition of Polynomial Degree

First, it is important to understand what a polynomial is. A polynomial is an expression that involves variables and coefficients. The 'degree' of this polynomial is the largest sum of the exponents of the variables in a single term within the polynomial. In other words, it is the highest power of the variable in the polynomial.
02

Identify Each Term of the Polynomial

Next, break down the polynomial into its individual terms. A term is a separate part of the polynomial and is typically separated by a '+' or a '-' sign. In two-variable polynomials, a term usually has both variables. For example, in the polynomial \(3x^2y + x^3 + 4xy^2\), \(3x^2y\), \(x^3\), and \(4xy^2\) are the terms.
03

Calculate Degree of Each Term

For each term, add up the exponents of the variables, which is also referred to as the 'degree' of the term. For instance, the term \(3x^2y\) has a degree of 3 (2 from \(x^2\) and 1 from \(y\)). Similarly, the term \(x^3\) has a degree of 3, and the term \(4xy^2\) has a degree of 3 (1 from \(x\) and 2 from \(y^2\)).
04

Identify the Highest Degree

After calculating the degree of each term, identify the term with the highest degree. This will give the degree of the polynomial. In our example, all the terms have the same degree (3), so the degree of the polynomial \(3x^2y + x^3 + 4xy^2\) is 3.

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Most popular questions from this chapter

Explain how to add or subtract rational expressions with the same denominators.

Perform the indicated operations. Simplify the result, if possible. $$\left(4-\frac{3}{x+2}\right)\left(1+\frac{5}{x-1}\right)$$

Perform the indicated operations. $$\left(1-\frac{1}{x}\right)\left(1-\frac{1}{x+1}\right)\left(1-\frac{1}{x+2}\right)\left(1-\frac{1}{x+3}\right)$$

Exercises \(142-144\) will help you prepare for the material covered in the next section. Simplify and express the answer in descending powers of \(x\) : $$ 2 x\left(x^{2}+4 x+5\right)+3\left(x^{2}+4 x+5\right) $$

If you are given a quadratic equation, how do you determine which method to use to solve it?
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