Problem 93 - Algebra A Combined Function

Question

Which of the following is/are factored form(s) of \(-2 x+14 ?\) a. \(-2(x+7)\) b. \(-2 \cdot x+14\) c. \(-2(x-14)\) d. \(-2(x-7)\)

Step-by-Step Solution

Answer

Option d: \(-2(x-7)\) is the factored form of \(-2x + 14\).

Step 1: Identify the Expression to Be Factored

The expression we need to factor is \(-2x + 14\). Our goal is to write this expression in factored form by finding a common factor.

Step 2: Determine the Greatest Common Factor (GCF)

Look at the terms \(-2x\) and \(14\). The greatest common factor is \(-2\), which is the largest number that divides both coefficients (-2 and 14) evenly.

Step 3: Factor Out the GCF

Factor out \(-2\) from the expression. Divide each term by \(-2\), which gives us:\[-2x + 14 = -2(x - 7)\].

Step 4: Check the Provided Options

Compare the options with the factored form:- **Option a:** \(-2(x+7)\) is not equivalent because after distribution it gives \(-2x - 14\).- **Option b:** \(-2 \cdot x + 14\) is not a completely factored form of \(-2x + 14\).- **Option c:** \(-2(x-14)\) would give \(-2x + 28\), which is not equivalent.- **Option d:** \(-2(x-7)\) matches our derived factored form and thus is equivalent.

Key Concepts

Greatest Common Factor (GCF)

The Greatest Common Factor, often abbreviated as GCF, is a foundational concept in algebra used to simplify expressions and equations. To find the GCF of algebraic terms, you look for the largest factor that divides each term evenly. This process requires you to examine both the coefficients (numerical part) and the variables. For example, if we take the expression

• \(-2x + 14\)

,we want to determine a common factor that can be factored out. The terms here are \(-2x\) and \(14\), and the greatest common factor is \(-2\). This is because \(-2\) is the largest number that can evenly divide all coefficients involved in the terms. Dividing both terms by \(-2\), you simplify the expression, ultimately changing its form to reveal underlying structures. It's a vital step because it sets the stage for further simplifying or solving equations.

Factored Form

Factored form in algebra refers to the expression of an equation as a product of its factors rather than a sum or difference of its terms. It provides a compact way to express complex algebraic expressions by highlighting their multiplicative structure. For instance:

• The expression \(-2x + 14\) can be rewritten by factoring out its greatest common factor.

When we factor out \(-2\), the expression becomes \(-2(x - 7)\). This transformation from the original expression to its factored form offers several advantages:

• It simplifies calculations and can make solving equations more straightforward.
• Factored forms often make it easier to identify roots or solutions if the expression is set to zero.

Understanding and transforming equations into their factored forms is a crucial algebraic skill that aids in simplifying, solving, and understanding mathematical relationships.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations that represent a particular value or set of values. They are fundamental in algebra, serving as the building blocks for equations and functions. Each expression can involve:

• Different operations such as addition, subtraction, multiplication, and division.
• Variables that stand for unknown values and coefficients which are the numerical factors of the variables.

For example, consider the expression \(-2x + 14\). Here:

• \(-2x\) is a term consisting of a coefficient \(-2\) and a variable \(x\).
• \(14\) is a constant term.

In algebra, manipulating algebraic expressions is essential to solve equations and inequalities, model real-world scenarios, and simplify complex problems. Through operations such as factorization, expressions can be rewritten, allowing for easier manipulation and understanding of the relationships they represent.