Problem 61 - College Algebra

Question

Perform the indicated operations Indicate the degree of the resulting polynomial. $$ \left(4 x^{2} y+8 x y+11\right)+\left(-2 x^{2} y+5 x y+2\right) $$

Step-by-Step Solution

Answer

The resulting polynomial from adding the given polynomials is \(2x^{2}y + 13xy + 13\). The degree of this polynomial is 3.

Step 1: Add the Polynomials

Combine the like terms from both polynomials. Like terms are those which have the same variables raised to the same power. That is, add \(4x^{2}y\) and \(-2x^{2}y\) together, and add \(8xy\) and \(5xy\) together. Leave \(11 + 2\) as they have no variables.

Step 2: Resultant Polynomial

Adding the terms from Step 1 will give the resulting polynomial: \( (4x^{2}y - 2x^{2}y) + (8xy + 5xy) + (11 + 2) = 2x^{2}y + 13xy + 13 \)

Step 3: Degree of the Polynomial

The degree of the polynomial is the highest power of x or y in the polynomial. For the resulting polynomial \(2x^{2}y + 13xy + 13\), the terms are of degree 3, 2, and 0, respectively. Therefore, the degree of the resulting polynomial is 3.

Key Concepts

Degree of a Polynomial

The degree of a polynomial is an important concept to grasp, as it helps determine the behavior and properties of the polynomial. The degree is defined as the highest power of the variable in the polynomial expression. In our given example of the resulting polynomial, \(2x^{2}y + 13xy + 13\), each term has its own degree.

• The term \(2x^{2}y\) is a product of \(x^2\) and \(y\), giving it a degree of 3 (as we add the exponents: 2 from \(x^2\) and 1 from \(y\)).
• The term \(13xy\) has a degree of 2 (1 for \(x\) and 1 for \(y\)).
• The constant term \(13\) holds a degree of 0 since it has no variable attached.

When identifying the degree of a polynomial, always look for the term with the highest total power of the variables involved. Here, the highest power is 3, so the degree of this polynomial is 3.

Like Terms

Understanding like terms is fundamental in simplifying polynomials. Like terms are terms in an expression that have the same variables and corresponding powers. They can be combined through addition or subtraction to simplify the expression.

In our expression \((4x^{2}y + 8xy + 11) + (-2x^{2}y + 5xy + 2)\), we identify the following like terms:

• \(4x^{2}y\) and \(-2x^{2}y\): Both terms have the same variables \(x^2y\).
• \(8xy\) and \(5xy\): These terms share the variables \(xy\) with the same power.
• Constants \(11\) and \(2\) can be added together without any variables being involved.

By recognizing and combining like terms, we simplify and manage polynomial expressions more effectively, leading to easier further algebraic manipulation.

Combining Polynomials

Combining polynomials involves carrying out addition or subtraction, which means merging their corresponding terms to get a simplified result. This process often requires dealing with like terms. There is also a systematic approach to ensure accuracy.

In the given exercise, we have two polynomials: \((4x^{2}y + 8xy + 11)\) and \((-2x^{2}y + 5xy + 2)\). To combine these polynomials, follow these steps:

• Arrange terms over each other, so like terms line up vertically.
• Add or subtract the coefficients of like terms.
• Write the resulting polynomial, ensuring every term is accounted for.

This process results in a single, unified polynomial, \(2x^{2}y + 13xy + 13\). Not only does this simplify
the expression, but it also brings out the overall structure of the polynomials involved, aiding in further operations and analysis.