Problem 3 - Algebra for College Students

Question

Determine the degree of the given polynomials. $$-x^{2} y+2 x y^{2}-x y$$

Step-by-Step Solution

Answer

The degree of the polynomial is 3.

Step 1: Understand the Problem

We need to determine the degree of a polynomial which is a measure of the highest power of the terms when the polynomial is expanded.

Step 2: Identify Each Term and Its Degree

The polynomial given is \(-x^2 y + 2xy^2 - xy\). Identify the degree of each term:1. \(-x^2 y\): The degree is the sum of the exponents, which is 3 (\(2 + 1\)). 2. \(2x y^2\): The degree is the sum of the exponents, which is 3 (\(1 + 2\)).3. \(-xy\): The degree here is 2 (\(1 + 1\)).

Step 3: Determine the Highest Degree

Compare the degrees of each term identified in Step 2. The terms are \(-x^2 y\) and \(2xy^2\) both with a degree of 3, and \(-xy\) with a degree of 2. The highest degree among these terms is 3.

Step 4: Conclude the Degree of the Polynomial

The degree of the polynomial, determined as the highest degree from the terms identified, is 3.

Key Concepts

Understanding Polynomial Terms

Polynomials are a sum of terms, and each term is made up of variables and coefficients. In the polynomial \(-x^2 y + 2xy^2 - xy\), there are several individual terms.

• The first term is \(-x^2 y\), which consists of the variable \(x\) raised to the 2nd power and \(y\) raised to the 1st power.
• The second term is \(2xy^2\), combining a coefficient 2 with \(x\) to the 1st power and \(y\) to the 2nd power.
• Lastly, \(-xy\) is a term where both \(x\) and \(y\) are to the 1st power.

Every term is essentially a product of a number (the coefficient) and variables raised to certain powers. Understanding each term correctly helps in calculating the degree effectively.

Calculating the Degree of a Polynomial

The degree of a polynomial is the largest sum of exponents in any term, which represents the highest power of that term. To solve this polynomial's degree:

• First, determine the degree of each term by adding the exponents in that term. For \(-x^2 y\), add 2 (from \(x\)) and 1 (from \(y\)) to get a degree of 3.
• Next, in \(2xy^2\) add 1 (from \(x\)) and 2 (from \(y\)) yielding a degree again of 3.
• For \(-xy\), both \(x\) and \(y\) contribute 1 to the degree, resulting in a degree of 2.

This calculation process involves breaking down each term into its simplest parts and then adding their exponents.

Exploring Exponents in Polynomials

Exponents indicate how many times a number or variable is multiplied by itself. In polynomials, understanding these exponents is crucial for finding the degree.

• For instance, in the term \(-x^2 y\), the exponent 2 tells us that \(x\) is squared, meaning \(x\) multiplied by \(x\).
• Similarly, the \(2xy^2\) term includes \(y^2\), implying \(y\) times \(y\).
• The more factors involved through exponents, the higher the term's power becomes, which is important in determining the polynomial's degree.

Recognizing these exponents helps to quickly evaluate the terms' degrees and ultimately find the overall degree of the polynomial.