Problem 34 - Elementary and Intermediate Algebra

Question

Add the polynomials. $$ (-0.3 r-5.2 s)+(0.8 r-5.2 s) $$

Step-by-Step Solution

Answer

The sum of the polynomials is \(0.5r - 10.4s\).

Step 1: Identify Like Terms

In the expression \((-0.3 r - 5.2 s) + (0.8 r - 5.2 s)\), identify the like terms, which are the terms with the same variable part. Here, \(-0.3r\) and \(0.8r\) are like terms as they both contain \(r\). Similarly, \(-5.2s\) is a like term with itself.

Step 2: Add the "r" Terms

Add the coefficients of the \(r\) terms together. Combine \(-0.3r + 0.8r\). To do this, calculate the sum of the coefficients: \(-0.3 + 0.8 = 0.5\). This gives us \(0.5r\).

Step 3: Add the "s" Terms

Now add the \(s\) terms. Since both are \(-5.2s\), their sum is \(-5.2s + (-5.2s) = -10.4s\).

Step 4: Combine the Results

Combine the results from Step 2 and Step 3 to form the final expression: \(0.5r - 10.4s\).

Key Concepts

Understanding Like Terms

In polynomial expressions, like terms are terms that have the same variable raised to the same power. They are crucial when adding or subtracting polynomials because you can only combine like terms.

• For example, in the expression \(-0.3r\) and \(0.8r\), both terms have the variable \(r\) raised to the 1st power, making them like terms.
• Similarly, terms like \(-5.2s\) in our example are like terms with each other because they contain the variable \(s\) with the same degree (which is 1).

When adding or subtracting polynomials, it's important to first identify like terms and then combine them by adding or subtracting their coefficients. This helps simplify the polynomial to its lowest form. Identifying like terms makes it easier to handle complex expressions, ensuring each variable component is combined correctly with its respective part.

The Role of Coefficients

Coefficients are numerical factors that multiply the variable part of a term in an algebraic expression. They are used to show how many times a variable is being considered in the expression.

• For instance, in the term \(-0.3r\), \(-0.3\) is the coefficient that multiplies the variable \(r\).
• In the problem, both \(-0.3r\) and \(0.8r\) have coefficients which can be added: \(-0.3 + 0.8 = 0.5\).

When solving polynomial addition problems, you only combine coefficients of like terms. It's essential to work accurately with these numbers to ensure the final expression is correct. The coefficients tell you the magnitude of the variable's impact in that part of the equation, allowing us to effectively manage and simplify the expressions.

Explaining Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and mathematical operations. It does not contain an equality sign and represents a value.

• In our example, the expression \((-0.3r - 5.2s) + (0.8r - 5.2s)\) consists of terms that need to be combined.
• Understanding the structure of an algebraic expression is critical in manipulating and simplifying it, as well as solving related algebra problems.

An algebraic expression is composed of terms, each comprising a coefficient and a variable part. Familiarizing yourself with these aspects allows for smoother manipulation, enabling the solving of even more complex equations over time. Expressions form the building blocks of equations and inequalities, which are core elements of algebra and critical in various applications in mathematics.