Problem 60 - College Algebra

Question

$$\begin{aligned}&\text , \text { Perform the indicated operations. Indicate}\\\ &\text { the degree of the resulting polynomial.}\end{aligned}$$ $$\left(-2 x^{2} y+x y\right)+\left(4 x^{2} y+7 x y\right)$$

Step-by-Step Solution

Answer

The resulting polynomial is \(2x^{2}y + 8xy\) and its degree is 3.

Step 1 - Identify like terms

The like terms in the given equations are \(-2x^{2}y\) and \(4x^{2}y\), \(xy\) and \(7xy\). Now the like terms can be clubbed together.

Step 2 - Perform the operations

Add the coefficients of the like terms. That is, \(-2x^{2}y + 4x^{2}y = 2x^{2}y\), and \(xy + 7xy = 8xy\)

Step 3 - Write the resulting polynomial

Our resulting polynomial is \(2x^{2}y + 8xy\)

Step 4 - Indicate the degree of the resulting polynomial

The degree of the polynomial is the highest sum of the exponents of \(x\) and \(y\). This happens in \(2x^{2}y\), for which the degree is \(2+1=3\). Hence, the degree of the resulting polynomial is 3.

Key Concepts

Combining Like Terms

Understanding how to combine like terms is essential when you want to simplify algebraic expressions, including polynomials. Like terms are terms within an expression that have exactly the same variables raised to the same powers. For example, in the terms \( -2x^{2}y \) and \( 4x^{2}y \) the variables \( x \) and \( y \) are present in both and are raised to the same power, namely 2 for \( x \) and 1 for \( y \), making them like terms.

When combining like terms, you simply add or subtract the coefficients (the numerical parts) of these terms while keeping the variable part unchanged. So, in the case of our example, combining \( -2x^{2}y \) and \( 4x^{2}y \) gives us \( ( -2 + 4 )x^{2}y \) which simplifies to \( 2x^{2}y \). It's important not to mix up different terms; only those exactly matching in variables and their exponents can be combined.

Polynomial Degree

Every polynomial has a degree, which is a critical concept in algebra. The degree of a polynomial is the highest sum of exponents of the variables in a single term of the polynomial. In simpler terms, it's the largest combined power to which the variables are raised.

For instance, if we look at our resulting polynomial \( 2x^{2}y + 8xy \), we have to identify the term with the highest degree. The term \( 2x^{2}y \) has the variables \( x \) and \( y \) with the exponents 2 and 1, respectively. Adding these exponents gives us 3, which means that the degree of this term is 3. Since no other term in the polynomial has a higher degree, the polynomial itself has a degree of 3. It’s important to note that the degree of a polynomial dictates the general shape of its graph. For example, a third-degree polynomial will generally have at least one point of inflection.

Adding Polynomials

Adding polynomials is a pretty straightforward operation once you know how to identify and combine like terms. To add polynomials, simply arrange each polynomial with like terms aligned vertically, and then add the coefficients of like terms to get the sum.

In the exercise, we added the polynomials \( -2x^{2}y+x y \) and \( 4x^{2}y+7 x y \) by combining like terms. We lined up \( -2x^{2}y \) with \( 4x^{2}y \) and \( xy \) with \( 7xy \), and then added their coefficients to find \( 2x^{2}y + 8xy \) as the resulting sum of the polynomials. This process is the same no matter how many terms are present or how complicated the polynomials are. Just keep in mind that you can only add the coefficients of like terms and that the variable parts remain unchanged in the sum.