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Chapter 5: Problem 89

Add. $$ \left(8 x^{2} y-3 x y^{2}+4 x y\right)+\left(-2 x^{2} y-x y^{2}+x y\right) $$

Short Answer

Expert verified
The simplified expression is \(6x^2y - 4xy^2 + 5xy\).

Step by step solution

01

- Identify like terms

First, identify the like terms in the two expressions. Like terms are terms that have the same variables raised to the same powers. Here, the like terms are: 1. The terms with the variable combination \(x^2 y\): \(8x^2y\) and \(-2x^2y\).2. The terms with the variable combination \(xy^2\): \(-3xy^2\) and \(-xy^2\).3. The terms with the variable combination \(xy\): \(4xy\) and \(xy\).
02

- Add the coefficients of like terms

Add the coefficients of the like terms: 1. For \(x^2 y\): \(8x^2y + (-2x^2y) = (8-2)x^2y = 6x^2y\).2. For \(xy^2\): \(-3xy^2 + (-xy^2) = -3xy^2 - xy^2 = -4xy^2\).3. For \(xy\): \(4xy + xy = 4xy + 1xy = 5xy\).
03

- Combine the results

Combine the results from step 2 to form the final expression: \(6x^2y - 4xy^2 + 5xy\). This is the simplified form of the original expression.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

like terms
When dealing with polynomial expressions, recognizing 'like terms' is a fundamental skill. Like terms are terms that contain the same variables raised to the same powers. Identifying them is crucial because it'll help you to combine and simplify the expression. For example, in the polynomial expressions given, terms with the variable combination \( x^2 y \) are like terms. Similarly, terms with \( xy^2 \) and \( xy \) are also like terms. Ensuring you correctly identify these will make the process of addition much smoother and error-free.
coefficients
Coefficients are the numerical part of a term that multiplies the variable(s). In the polynomial expression \( 8x^2y \), the coefficient is 8. When adding polynomials, you focus on the coefficients of like terms. For instance, if we consider the terms \( 8x^2y \) and \( -2x^2y \), we simply add their coefficients, which are 8 and -2, to get 6. This results in the term \( 6x^2y \). Understanding the role coefficients play is essential for accurate manipulation of polynomial expressions.
simplifying expressions
Simplifying expressions involves combining like terms to reduce the polynomial to its simplest form. This means you only need to perform operations on the coefficients of like terms. For example, after identifying and adding the like terms in \( (8x^2y - 3xy^2 + 4xy) + (-2x^2y - xy^2 + xy) \), we simplify it to \( 6x^2y - 4xy^2 + 5xy \). By doing so, the expression is easier to work with, and it gives a clearer, consolidated view of all the terms involved.
polynomial expressions
Polynomial expressions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. In the given problem, we added two polynomial expressions. The resultant polynomial still maintains the characteristics of having variables with powers and coefficients: \( 6x^2y - 4xy^2 + 5xy \). Understanding how to manipulate polynomial expressions, including identifying like terms and simplifying, forms the basis for more complex algebraic operations.

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