Problem 59 - Algebra and Trigonometry

Question

In Exercises 59–66, 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 sum of the polynomials \(5x^{2}y -3xy\), and \(2x^{2}y - xy\) is \(7x^{2}y - 4xy\). The degree of the resulting polynomial is 3.

Step 1: Identify Like Terms

Like terms are terms with the exact same variables raised to the exact same powers. Here, we observe that the terms \(5x^{2}y\) and \(2x^{2}y\) are like terms. Similarly, \(-3xy\) and \(-xy\) are like terms.

Step 2: Add Like Terms

Add the coefficients (numbers in front of the variables) of the like terms. Adding coefficients of \(5x^{2}y\) and \(2x^{2}y\) we get \(5 + 2 = 7\), and for \(-3xy\) and \(-xy\) we get \(-3 - 1 = -4\). The resulting polynomial is \(7x^{2}y - 4xy\).

Step 3: Determine Degree of the Polynomial

The degree of a polynomial is determined by adding the powers of the variables in each term and selecting the highest sum. The term \(7x^{2}y\) has degree 3 (2 from \(x^{2}\) and 1 from \(y\)), and the term \(-4xy\) has degree 2 (1 from \(x\) and 1 from \(y\)). Therefore, the degree of the resulting polynomial \(7x^{2}y - 4xy\) is 3 as that is the highest degree among the terms.

Key Concepts

Like Terms in Polynomials

Understanding like terms in polynomials is essential for simplifying expressions. Like terms are terms that have the exact same variables raised to the same powers. Essentially, they "look alike" in the context of the variables and their exponents, although they may have different coefficients (the numbers in front).

• Example: In the expression \(5x^2y - 3xy + 2x^2y\), both \(5x^2y\) and \(2x^2y\) are like terms because they share the variable arrangement \(x^2y\).
• Also, in the same expression, \(-3xy\) is like with \(-xy\) as both have the variables \(xy\).

Identifying like terms allows you to combine them to simplify the polynomial expression. This is a fundamental skill when working on polynomial operations.

Adding Polynomials

Once like terms are identified, the next step in polynomial operations is to add them. This involves combining the coefficients of like terms while keeping the variable parts unchanged.

• For instance, when adding \(5x^2y\) and \(2x^2y\), simply add their coefficients: \(5 + 2 = 7\). The result is \(7x^2y\).
• Similarly, for \(-3xy\) and \(-xy\), adding the coefficients gives \(-3 - 1 = -4\), leading to the term \(-4xy\).

Thus, the sum of the polynomials \((5x^2y - 3xy) + (2x^2y - xy)\) results in \(7x^2y - 4xy\). Adding polynomials is mainly about adding these coefficients while respecting the form of the terms.

Degree of a Polynomial

The degree of a polynomial is an important concept that tells you the highest power of any individual term in the polynomial. It helps in understanding the complexity and behavior of the polynomial.- To find out the degree, you need to look at each term. For a term like \(x^m y^n\), the sum of the exponents \(m+n\) provides the degree of that term.- Consider the polynomial \(7x^2y - 4xy\): - The term \(7x^2y\) has the degree \(2 + 1 = 3\). - The term \(-4xy\) has the degree \(1 + 1 = 2\).Therefore, the degree of the entire polynomial is the highest of these, which is 3. Knowing the degree helps you understand the polynomial's behavior, especially when graphing it or determining the number of possible roots.