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Chapter 5: Problem 53

\(\left(7 x^{2}-2 x y+6 y^{2}\right)+\left(3 x^{2}-5 x y\right)\)

Short Answer

Expert verified
10x^{2} - 7xy + 6y^{2}

Step by step solution

01

- Understand the Expression

The given expression is \( (7x^{2} - 2xy + 6y^{2}) + (3x^{2} - 5xy) \). This expression is a sum of two polynomials.
02

- Group Like Terms

Combine the like terms from both polynomials: \(7x^{2}\text{ and } 3x^{2},\text{ } -2xy\text{ and } -5xy,\text{ and } 6y^{2}\text{ and } 0\text{ (no } y^{2}\text{ term in the second polynomial)} \).
03

- Combine Like Terms

Add the coefficients of like terms: \(7x^{2} + 3x^{2} = 10x^{2}\ -2xy - 5xy = -7xy\ 6y^{2} + 0 = 6y^{2} \).
04

- Write the Simplified Expression

Combine the results from the previous step: \( 10x^{2} - 7xy + 6y^{2} \).

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Key Concepts

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

Like Terms
Like terms are terms that have identical variable parts with the same exponents. For instance, in the expression given above, we have terms like \(7x^{2} \text{ and } 3x^{2}\) as like terms because both have \(x^{2}\) as the variable part. Similarly, \(-2xy \text{ and } -5xy\) are like terms due to having the same variable part \(xy\).
Identifying like terms is essential in polynomial addition because it allows us to combine them, thus simplifying the expression. When grouping like terms, ensure that the variables and their respective exponents match exactly.
Coefficients
Coefficients are the numerical multipliers of the terms in a polynomial. In the expression \(7x^{2} - 2xy + 6y^{2} + 3x^{2} - 5xy\), the coefficients are the numbers preceding the variables:
  • 7 in \(7x^{2}\)
  • -2 in \(-2xy\)
  • 6 in \(6y^{2}\)
  • 3 in \(3x^{2}\)
  • -5 in \(-5xy\)

When adding polynomials, combine the coefficients of the like terms. For example, adding \(7x^{2} \text{ and } 3x^{2}\) results in \(10x^{2}\) because \(7 + 3 = 10\). The coefficients dictate how many units of each term are present and must be carefully managed during math operations.
Simplification of Expressions
Simplification of expressions involves combining like terms and ensuring that the polynomial is in its simplest form. This process can be broken down into clear steps:
  • Identify like terms within the polynomial.
  • Combine the coefficients of like terms.
  • Rewrite the expression with the combined terms.

In this exercise, we started with \( (7x^{2} - 2xy + 6y^{2}) + (3x^{2} - 5xy) \). By identifying the like terms \(7x^{2}\text{ and } 3x^{2}, -2xy\text{ and } -5xy\), and combining their coefficients, we get the simplified form as \(10x^{2} - 7xy + 6y^{2}\). Simplification not only makes the expression easier to understand but also prepares it for further operations or evaluations.

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