Problem 56 - Algebra for College Students

Question

Perform the indicated operations. When possible write down only the answer. $$\left(x^{2}+y^{2}\right) \div(-1)$$

Step-by-Step Solution

Answer

-x^{2} - y^{2}

Step 1: Understand the Operation

Identify the operation required. In this case, you need to divide the expression \( x^{2} + y^{2} \) by -1.

Step 2: Apply the Division

Divide each term in the expression \( x^{2} + y^{2} \) by -1. This gives us \( \frac{x^{2}}{-1} + \frac{y^{2}}{-1} \).

Step 3: Simplify the Expression

Simplify the division for each term. \( \frac{x^{2}}{-1} = -x^{2} \) and \( \frac{y^{2}}{-1} = -y^{2} \).

Step 4: Combine the Simplified Terms

Combine the simplified terms to get the final result: \( -x^{2} - y^{2} \).

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables (like x and y), and operation symbols (like +, -, *, and /).
They do not contain an equal sign (=).
In this exercise, the algebraic expression is \(x^{2} + y^{2}\).
Here, \(x^{2}\) and \(y^{2}\) are terms that include variables raised to the power of 2.
Understanding different elements within algebraic expressions is crucial for performing operations like division and simplification.
Each term can be manipulated individually, making it easier to apply operations and simplify.

Division Operations

Division operations in algebra involve dividing one algebraic expression by another.
In this case, we are dividing the expression \(x^{2} + y^{2}\) by -1.
To do this, we need to divide each term of the expression by -1 separately:

• Divide \(x^{2}\) by -1: \(\frac{x^{2}}{-1}\)
• Divide \(y^{2}\) by -1: \(\frac{y^{2}}{-1}\)

Each divided term needs to be simplified to get a better understanding of the final result.
This leads us to the next key concept, simplifying polynomials.

Simplifying Polynomials

Simplifying polynomials means making the polynomial expression as simple as possible.
We start by simplifying each term from the division step:

• \(\frac{x^{2}}{-1} = -x^{2}\)
• \(\frac{y^{2}}{-1} = -y^{2}\)

After simplifying, we combine the terms to write them in a simple and concise way:
\(-x^{2} - y^{2}\)
This final expression is the simplest form of \(x^{2} + y^{2}\) divided by -1.
Simplifying polynomials helps in understanding the structure and properties of the equation more clearly.
Remember to always check each step carefully to ensure accuracy.