Problem 59 - College Algebra

Question

Perform the indicated operations Indicate the degree of the resulting polynomial. $$ \left(5 x^{2} y-3 x y\right)+\left(2 x^{2} y-x y\right) $$

Step-by-Step Solution

Answer

The result of the operation is \(7 x^{2} y - 4 x y\) and the degree of the resulting polynomial is 3.

Step 1: Combine like terms

Start by identifying and combining like terms in the given equation. Basically, \(5 x^{2} y\) and \(2 x^{2} y\) can be added together as they are terms of the same type \(x^{2}y\), likewise \(-3 x y\) and \(- x y\). Thus, the equation now simplifies to:\[(5 x^{2} y + 2 x^{2} y) + (-3 x y - x y)\]

Step 2: Perform the addition

This step involves actually adding the coefficients of the like terms as identified above. Also, carry each term’s variable along while performing this operation. The equation now becomes:\[7 x^{2} y - 4 x y\]

Step 3: Identify the degree of the polynomial

The degree is considered as the maximum power of x or y in the polynomial. In our case, there are two terms with degrees 2 and 1 respectively. The highest degree is 2 as displayed in the term \(7 x^{2} y\). The variable x is raised to the power of 2 and y is raised to the power of 1. So the maximum power \(2 + 1 = 3\), thus the degree of the polynomial is 3.

Key Concepts

Combining Like Terms

Combining like terms is crucial for simplifying polynomials effectively. Like terms in a polynomial are terms that have identical variable parts, meaning the same variables raised to the same powers. The key to combining them is to smoothly add or subtract their coefficients, keeping the variable parts intact.
For instance, in the expression given in the exercise \[ (5 x^{2} y - 3 x y) + (2 x^{2} y - x y) \]notice how terms like \(5x^2y\) and \(2x^2y\) are like terms because they both include \(x^2y\). Similarly, \(-3xy\) and \(-xy\) are like terms.

• Start by arranging like terms side by side.
• Add the coefficients: \(5 + 2\) gives \(7x^2y\) and \(-3 - 1\) gives \(-4xy\).

Combining like terms simplifies the polynomial, making it easier to handle in future calculations or operations.

Degree of a Polynomial

The degree of a polynomial provides valuable information about the polynomial's power and complexity. It is determined by the term with the highest total sum of exponents of its variables.
For a single term, the degree is the sum of the exponents of the variables in that term. In a full polynomial, the degree is the greatest degree among its terms.
Consider our simplified result from the exercise: \[7x^2y - 4xy\]

• The term \(7x^2y\) has degree 3 because \(2 + 1 = 3\).
• The term \(-4xy\) has degree 2 because \(1 + 1 = 2\).

The highest degree among these terms is 3, hence the polynomial's degree is 3. The degree signifies how the variables in the polynomial are interacting with each other at the highest level.

Addition of Polynomials

Adding polynomials involves performing arithmetic addition on polynomials' like terms. This process is straightforward yet requires careful alignment of terms to prevent errors, primarily focusing on combining like terms. When you're ready to add, ensure all terms with the same variable powers are properly aligned.
Adding polynomials is often represented as a vertical or horizontal addition. Here’s the approach used in our exercise:

• Line up the polynomials ensuring the like terms are positioned together.
• Add (or subtract) their numerical coefficients.

For instance:\[(5x^2y - 3xy) + (2x^2y - xy)\]By adding the like terms, we result in the expression:\[7x^2y - 4xy\]This technique ensures that each group of like terms is simplified correctly, producing an accurate and manageable polynomial as the result.