Problem 69 - Elementary Algebra

Question

For the following problems, a term will be given followed by a group of its factors. List the coefficient of the given group of factors. $$ 5 a ; 5 $$

Step-by-Step Solution

Answer

Answer: 1

Step 1: Identify the term and the group of factors

The given term is \(5a\), and the group of factors is \(5\).

Step 2: Divide the term by the group of factors

To find the coefficient, we need to divide the term (\(5a\)) by the group of factors (\(5\)). We do this by dividing the numbers and the variables separately: \( \frac{5a}{5} = \frac{5}{5} \cdot a \)

Step 3: Simplify the division

Now, we need to simplify the fraction: \( \frac{5}{5} \cdot a = 1 \cdot a \)

Step 4: Write the coefficient

The coefficient of the group of factors \(5\) in the term \(5a\) is \(1\). Thus, the answer is \(1\).

Key Concepts

Factorization

Factorization in algebra is the process of breaking down expressions into groups of simpler elements, called factors, that when multiplied together give the original expression. Understanding factorization is crucial because it helps in simplifying expressions and solving equations efficiently.

When you factorize an expression, you're essentially finding what elements multiply together to create it:

• For numbers, think about breaking them down into prime factors (e.g., 12 into 2 and 3).
• For algebraic expressions, it means expressing them as a product of their simplest components, like turning a polynomial into a product of monomials or smaller polynomials.

In the context of the exercise, identifying and dividing by the factor is the step that involves factorization. By recognizing that the term "5" is a factor of "5a," we performed a division to further simplify it into "1" (the coefficient) and "a" (the variable). Understanding this practice allows for easier manipulation and simplification of more complex algebraic terms.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They form the foundation for understanding and solving problems in algebra. Expressions can range from simple ones, such as \(3x + 4\), to complex ones involving multiple terms and operations.

An algebraic expression comprises:

• Terms, which are the individual components separated by plus or minus signs.
• Coefficients, which are the numbers multiplied by variables.
• Variables, which are symbols (like \(x\) or \(a\)) that stand in for unknown values.
• Constants, which are numbers without variables attached.

The term \(5a\) from the exercise is an algebraic expression with one term. It consists of the coefficient \(5\) and the variable \(a\). Understanding this structure helps in identifying parts that can be manipulated for solving, simplifying, or evaluating expressions in various mathematical contexts.

Simplification

Simplification is a critical skill in algebra that involves transforming expressions into their simplest form. The goal is to make the expressions easier to work with while ensuring they remain equivalent to their original form.

Simplification involves:

• Combining like terms, which means grouping similar variables and constants together.
• Performing basic arithmetic operations where applicable.
• Canceling common factors when dividing terms, as done in the provided solution.

In the example of \(5a\), simplification was achieved by factoring out and canceling the common factor "5," resulting in \(1 \cdot a\). This process not only made the coefficient of the factorless encumbrance easy to identify but also demonstrated a fundamental strategy to handle more complicated expressions effortlessly. Mastering simplification equips students with techniques to efficiently solve equations and cleanly present solutions.