Problem 60 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify Like Terms

In both polynomials, the terms \( -2x^{2}y \) and \( 4x^{2}y \) are alike since they share the same variables with the same exponents. The same applies to \( xy \) and \( 7xy \).

Step 2: Add Like Terms Together

Add the coefficients of the like terms together. This results in \( (-2+4)x^{2}y + (1+7)xy \), which simplifies to \( 2x^{2}y + 8xy \).

Step 3: Determine the Degree of the Polynomial

The degree of a term is the sum of the exponents of its variables, and the degree of a polynomial is the highest degree among all terms. Here, the first term, \( 2x^{2}y \), has a degree of 3 (2 from the \( x \) and 1 from the \( y \)), and the second term, \( 8xy \), has a degree of 2 (1 each from \( x \) and \( y \)). Thus, the degree of the polynomial is 3, which is the highest degree.

Key Concepts

Like Terms in Polynomials

Understanding like terms in polynomials is vital when dealing with polynomial operations. Like terms are terms that have exactly the same variables raised to the same powers, although coefficients may differ. For example, in terms with variables like \(x^2y\), all other \(x^2y\) terms are considered 'like terms', regardless of their coefficients. Imagine you have a basket of fruit; like terms would be all the apples grouped together, even if some are green and others are red.

Identifying like terms allows us to combine or operate on them. When combining like terms, we only combine the coefficients and leave the variable parts unchanged. This is similar to saying, 'if I have 2 apples and you give me 3 more, now I have 5 apples.' We are only adding up how many apples (or in our case, the coefficient of the terms) we have, not changing what fruit it is.

Adding Polynomials

Adding polynomials is similar to stacking blocks where each block has a label. When we add polynomials, we look for blocks with the same label (the like terms we identified) and place them together. This operation involves combining the coefficients of the like terms while keeping the variable part of the term intact. For example, with \( -2x^{2}y \) and \( 4x^{2}y \) from our exercise, we identify \( -2 \) and \( 4 \) as coefficients to combine.

In the given example, by adding \( -2x^{2}y + 4x^{2}y \) and \( xy + 7xy \) the like terms are combined to form \( 2x^{2}y + 8xy \). It's important to make sure that we only add numbers with numbers and variables with the same type of variables. Think of it like sorting and summing coins; only the same denomination coins (like quarters with quarters, dimes with dimes) can be directly summed up.

Degree of a Polynomial

The degree of a polynomial is the highest degree of its terms when the polynomial is expressed in standard form (terms written from highest to lowest degree). To find the degree, look at each term individually first. The degree of any single term is the sum of the exponents of the variables in it. For instance, in \( x^{2}y \) the degree is 3 (2 from \( x \) and 1 from \( y \) ).

After determining the degree of individual terms, compare them to find the highest one, which represents the degree of the entire polynomial. In our exercise, \( 2x^{2}y \) has a degree of 3 and is the term with the highest degree, so the polynomial \( 2x^{2}y + 8xy \) has a degree of 3. Knowing the degree of a polynomial is essential, as it gives us information about the polynomial's behavior, especially for graphing and solving polynomial functions. It tells us, in part, how many roots, or solutions, the polynomial might have, and the shape of its graph.