Problem 46 - Beginning and Intermediate Algebra

Question

Factor out the greatest common factor. Be sure to check your answer. Factor out \(-7\) from \(-14 k+21\)

Step-by-Step Solution

Answer

The factored form of the given expression \(-14 k+21\) is \(-7(2k - 3)\).

Step 1: Identify the greatest common factor

To start, we need to find the greatest common factor between the coefficients of the two terms. In our case, the coefficients are -14 and 21. The greatest common factor between these two numbers is 7. However, since our question asks us to factor out a negative value, we will factor out a -7 instead.

Step 2: Apply the Distributive Property

Now that we have identified the GCF as -7, we can apply the distributive property to factor it out of the entire expression. \(-7(2k - 3)\) Since we have factored out a -7, the equation now has two terms: 2k and -3.

Step 3: Check your work

Lastly, we need to check our work by distributing the -7 back into the expression to see if we get the original expression: \(-7(2k-3) = -14k + 21\) The expression checks out, so the factored form of the given expression is: \(-7(2k - 3)\)

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is the largest number that can evenly divide each term of an algebraic expression. When factoring, the goal is to simplify expressions by the largest factor that terms have in common. In the case of our exercise, the terms were

• \(-14k\)
• \(21\)

The coefficients here are \(-14\) and \(21\), and their greatest common factor is \(7\). Since the exercise specifies factoring out a negative value, we use \(-7\) as the GCF. This makes the expression cleaner and follows the exercise guidelines.
By knowing how to identify the greatest common factor, you will be better able to simplify algebraic expressions involving larger numbers.
Remember that the GCF helps to declutter terms, making the next steps in the solution straightforward.

Distributive Property

The distributive property is fundamental in algebra, allowing us to multiply a term across a sum or difference inside parentheses. It is described by the expression:

• \(a(b + c) = ab + ac\)

In the context of our problem, once the GCF is determined as \(-7\), we apply the distributive property to factor it from the expression. How was this done? Here's the breakdown:
We take \(-7\) and multiply it by each term inside the new parentheses:

• The first term is \(-7 \times 2k\), resulting in \(-14k\)
• The second term is \(-7 \times (-3)\), resulting in \(21\)

This means that the original expression \(-14k + 21\) becomes \(-7(2k - 3)\) when factored. The distributive property ensures that when you apply it, the operation is reversed correctly, giving us the original expression back upon distribution.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and arithmetic operations. When working with algebraic expressions like \(-14k + 21\), factoring plays a critical role in simplifying them. Each part of the expression serves a function:

• \(-14k\) is a term involving the variable \(k\)
• \(21\) is a constant term

Factoring these expressions can reduce complexity, making it easier to solve, graph, or interpret them in equations.
The prime objective is to express the terms in a simpler form using common factors or identities like the distributive property.
Factoring can also be vital in finding zeros of functions because it allows an expression to be expressed as a product of factors that can be set to zero in an equation. Understanding how to manipulate algebraic expressions reliably is key to mastering algebra and extending your math skills to more advanced topics.