Problem 97 - Algebra A Combined Function

Question

Which of the following expressions are factored? $$ 5(2 y+z)-b(2 y+z) $$

Step-by-Step Solution

Answer

The expression is factored as \((2y + z)(5-b)\).

Step 1: Understand the Expression

The given expression is \( 5(2y + z) - b(2y + z) \). We need to determine if this expression is factored, which means it is expressed as a product of its factors.

Step 2: Identify Common Factors

Observe that both terms in the expression, \( 5(2y + z) \) and \( -b(2y + z) \), contain the common factor \( (2y + z) \). This indicates that the terms share a common factor.

Step 3: Factor Out the Common Factor

Factor out \( (2y + z) \) from each term. This results in the expression \( (2y + z)(5 - b) \). Thus, the expression is written as a product of its factors, which satisfies the definition of a factored expression.

Key Concepts

Common Factors

In mathematics, a common factor is a value or expression that divides two or more numbers or algebraic terms without leaving a remainder. Identifying common factors is a key step in simplifying expressions. For example, consider the expression \(5(2y + z) - b(2y + z)\). Both terms share the common factor \((2y + z)\). This means that \((2y + z)\) is a common factor for the given expression. Identifying common factors can simplify complex expressions, making them easier to work with.

• Helps in breaking down complicated expressions.
• Useful in solving equations where simplifying expressions can aid comprehension.
• Allows us to see the structure of expressions more clearly.

Paying attention to common factors can save time and increase accuracy in solving algebraic problems.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operators (such as addition and multiplication). These expressions represent values and operations on these values. In the given example, \(5(2y + z) - b(2y + z)\), we see the use of multiplication and subtraction. The terms in an algebraic expression can be constants like \(5\), variables like \(y\) and \(z\), or products of these elements.

• Can be simple or complex based on the number of terms and operations involved.
• Model real-world situations to understand various relationships.
• Are the foundation for equations, which are statements of equality between expressions.

Working with algebraic expressions is essential as they enable us to generalize mathematical operations, helping to solve a wide variety of problems.

Factored Form

Factoring is the process of breaking down an expression into simpler components, called factors, that when multiplied together give back the original expression. An expression is said to be in factored form when it is written as the product of these factors. For the expression \(5(2y + z) - b(2y + z)\), factoring out the common term \((2y + z)\) results in the expression \((2y + z)(5 - b)\). This expression is now in its factored form, illustrating the clear separation into two distinct multiplicative components.

• Offers a cleaner and simplified view of the expression's structure.
• Makes solving equations easier, as factored expressions reveal roots and solutions.
• Helps in identifying constraints or limits by examining factors individually.

Understanding how to factor expressions is a crucial algebraic skill that simplifies problem-solving and aids in finding solutions more efficiently.