Problem 59 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify Like Terms

The like terms in the two polyomials are \(5x^{2}y\) and \(2x^{2}y\) then \(-3xy\) and \(-xy\).

Step 2: Add the Like Terms

Add the like terms together. The sum of \(5x^{2}y\) and \(2x^{2}y\) is \(7x^{2}y\). The sum of \(-3xy\) and \(-xy\) is \(-4xy\).

Step 3: State the Resulting Polynomial

After adding the like terms, the resulting polynomial is \(7x^{2}y - 4xy\).

Step 4: Determine the Degree of the Polynomial

The degree of a polynomial is determined by the highest sum of the exponents of each term. In the resulting polynomial, the term with the highest sum of exponents is \(7x^{2}y\), which has a sum of exponents equal to 3. Therefore, the degree of the polynomial is 3.

Key Concepts

Adding Polynomials

Performing operations on polynomials, particularly addition, is fundamental in algebra. When we add polynomials, we combine like terms to simplify the expression. Like terms are terms that have identical variable parts, meaning they have the same variables raised to the same powers. For example, in the expression \(5x^{2}y - 3xy + 2x^{2}y - xy\), the terms \(5x^{2}y\) and \(2x^{2}y\) are like terms, and so are \(3xy\) and \(xy\).

To add polynomials, we follow these steps:

• Align like terms.
• Add the coefficients of like terms.
• Write the sum with the combined like terms.

Using this method in our example, we get \(7x^{2}y - 4xy\) after combining \(5x^{2}y + 2x^{2}y\) and \(3xy + xy\). This action simplifies the polynomial and makes it easier to understand and work with in subsequent calculations.

Like Terms in Polynomials

Identifying like terms in polynomials is essential to simplifying expressions and performing operations correctly. Two terms are considered like terms if they have exactly the same variables to the same powers, regardless of the coefficients. For example, the terms \(3a^{2}b\) and \(5a^{2}b\) are like terms because they both contain the variables \(a\) and \(b\), each raised to the second and first power, respectively. However, \(3a^{2}b\) and \(3ab^{2}\) are not like terms because the powers of \(b\) are different.

In the given exercise, \(5x^{2}y\) and \(2x^{2}y\) are like terms, as are \(3xy\) and \(xy\). By understanding the concept of like terms, we can combine them effectively to simplify polynomial expressions, leading to efficient problem-solving techniques in algebra.

Degree of a Polynomial

The degree of a polynomial is a very important concept in algebra because it provides information about the polynomial's characteristics, including the number of solutions and the behavior of its graph. The degree is determined by the highest total power of any term in the polynomial when variables are considered.

To find the degree, look for the term with the highest sum of exponents on the variables. In the polynomial \(7x^{2}y - 4xy\), the term \(7x^{2}y\) has the highest degree because the sum of the exponents is 3 (\(2\) from \(x\) and \(1\) from \(y\)), making the polynomial a third-degree polynomial. Remember that constants, terms without variables, are considered to have a degree of zero. It’s also worth noting that the degree of a polynomial tells us about the polynomial's shape when graphed: the higher the degree, the more turns the graph can have.