Problem 52 - Beginning and Intermediate Algebra

Question

Factor out the greatest common factor. Be sure to check your answer. Factor out \(-1\) from \(-z-6\)

Step-by-Step Solution

Answer

The short answer is: \(-1(z + 6)\)

Step 1: Identify the Greatest Common Factor

In this case, the greatest common factor is given as -1.

Step 2: Divide each term by the GCF

Divide each term in the expression by -1: \(-\frac{z}{-1} = z\) \(-\frac{6}{-1} = 6\)

Step 3: Write the factored expression

The factored expression is the result of factoring out -1: \(-1(z + 6)\)

Step 4: Check your answer

To check if your answer is correct, distribute the -1 back into the expression: \(-1(z + 6) = -1 \times z + (-1) \times 6 = -z - 6\) Since the original expression is obtained, the factored expression is correct.

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is the largest number that can evenly divide all terms in an expression. In mathematics, this helps simplify expressions and solve equations more easily. To find the GCF:

• List the factors of each term in the expression.
• Identify the highest number that appears in each list of factors.

For an algebraic expression, the GCF might also include common variables.
In this case, the GCF for the expression \(-z-6\) is \(-1\). This GCF was chosen because it can evenly divide both terms of the expression without leaving any remainder. Factoring out the GCF simplifies the expression, making it easier to work with, especially in more complex algebraic manipulations.

Factoring Negative Numbers

Factoring negative numbers involves taking out \(-1\), so that any expression or polynomial becomes positive or has other desired properties. This can often help in aligning terms to make equations easier to solve or to match a particular format.
Consider the expression \(-z - 6\):

• Factoring out a \(-1\) allows us to write it as \(-1 \times (z + 6)\).

Mathematically, this is done by dividing each term by \(-1\), flipping their signs.
When you distribute back, multiplying each term by \(-1\), you should reach back to the original expression. This ensures accuracy in factoring. Factoring negatives is particularly useful to maintain consistency when dealing with equations and simplifying complex terms.

Algebraic Expressions

Algebraic expressions are combinations of coefficients, variables, and constants connected by operations like addition, subtraction, multiplication, and division.
These expressions can sometimes be complex, but factoring helps in breaking them down into simpler components.
Factoring out the GCF, as we did with \(-z - 6\), intends to simplify expressions, making them more manageable.
Here's a quick overview of some components:

• Variables: Symbols representing numbers, like \(z\).
• Coefficients: Numbers multiplied by variables, like the \(-1\) with \(z\).
• Constants: Standalone numbers like 6 in the expression.

Expressions like these require factoring for simplification or solving. Understanding how to manipulate and break down these elements enhances problem-solving skills in algebra.