Problem 76 - Elementary Algebra Within Reach

Question

$$ r^{2}+3 r s-6-r s $$

Step-by-Step Solution

Answer

The simplified expression of \(r^2 + 3rs -6 -rs\) is \(r^2 + 2rs - 6\)

Step 1: Identify the Like Terms

In the given expression, the like terms are \(3rs\) and \(-rs\), and these terms can be combined.

Step 2: Combine the Like Terms

When we combine \(3rs\) and \(-rs\), we get \(2rs\). So, the expression simplifies to \(r^2 + 2rs - 6\).

Step 3: Final Simplified Expression

The final simplified expression is \(r^2 + 2rs - 6\).

Key Concepts

Combining Like Terms

When working with algebraic expressions, combining like terms is a vital skill to simplify the expression effectively. Like terms are terms that contain the same variable raised to the same power. For instance, in the expression \(3rs\) and \(-rs\), these terms are like terms because both involve the variable products \(rs\). To simplify an expression:

• Identify terms that have the same variables and exponents.
• Add or subtract those coefficients.

In our example, we take \(3rs - rs\), where the like terms share the variable product \(rs\). By performing the arithmetic operation on their coefficients, we calculate \(3 - 1 = 2\). This results in the term \(2rs\), simplifying our original expression to \(r^2 + 2rs - 6\).
Combining like terms not only makes the expression simpler but also easier to work with in further calculations or problem-solving.

Algebra

Algebra is a branch of mathematics that helps us understand how to work with numbers and variables to solve equations and simplify expressions. It's like a puzzle, involving numbers, letters (or symbols), and operations such as addition, subtraction, multiplication, and division.Within algebra, each component of an expression can signify different properties:

• **Variables**: These are symbols like \(r\) and \(s\), used to represent unknowns or varying numbers.
• **Coefficients**: Numbers that stand in front of the variables, as seen in \(3\) in \(3rs\).
• **Constants**: Numbers on their own, such as \(-6\) in the expression.

Algebra gives us the tools to manipulate these elements to simplify or solve problems effectively. By understanding how each part of an expression works, you can apply strategies like combining like terms to manage complex equations.

Polynomials

Polynomials are a key concept in algebra, involving expressions that consist of sums and differences of terms. Each term is a product of a coefficient and a variable raised to a non-negative integer power. The given expression \(r^2 + 3rs - 6 - rs\) is an example of a polynomial.Here are some basic features of polynomials:

• **Degree of a Polynomial**: The highest exponent of the variable(s), which defines the degree. In \(r^2\), the degree is 2.
• **Standard Form**: Arranging the terms in descending order by their degrees, like \(r^2\), \(2rs\), and then the constant \(-6\).
• **Terms**: Each part of the polynomial separated by plus or minus signs. For example, \(r^2\), \(2rs\), and \(-6\) are distinct terms.

Understanding polynomials involves recognizing their structure so you can perform operations like addition, subtraction, and simplification. When combining like terms in a polynomial, we aim to make the expression less cumbersome while retaining its meaning. This lays the groundwork for more advanced algebraic concepts and solving polynomial equations.