91Ó°ÊÓ

In algebra, the **Greatest Common Factor (GCF)** is the largest factor that divides two or more numbers. When working with algebraic expressions, finding the GCF helps to simplify the problem. Remember:

• Factors are numbers that divide into another number without leaving a remainder.
• For terms with variables, the GCF includes the variable with the smallest power.

In our exercise, we need to find the GCF for the terms in the numerator: \(24h²k, 56hk², -28hk\). Here are the steps:

1. Identify the GCF for the coefficients (24, 56, -28). The GCF is 4.
2. For the variables, take the lowest power for common variables: \(h\) and \(k\). The GCF is \(hk\).

Combine the results: The GCF of these terms is \(4hk\). Once we factor out \(4hk\), the expression inside the parentheses is simpler.

Factoring breaks down expressions into products of simpler terms. It makes complex algebraic operations easier. To factor, follow these steps:

1. Identify the GCF of the terms.
2. Divide each term by the GCF and place the resulting expression inside parentheses.

For the exercise, after finding the GCF \(4hk\), we factor the expression: \[24h²k + 56hk² - 28hk = 4hk(6h + 14k - 7)\].

Notice how this simplifies the original problem, making it easier to work with.

Simplifying fractions often involves cancelling out common factors in the numerator and denominator. Here's how to approach it:

1. Write the fraction with the factored numerator.
2. Cancel out common factors present in both the numerator and the denominator.

Using our exercise, we rewrite it as: \[\frac{4hk(6h + 14k - 7)}{16h^2k^2}\].

The common factor \(4hk\) can be cancelled:
• Numerator: \[4hk(6h + 14k - 7)\]
• Denominator: \[16h^2k^2\] = \[4hk \times 4hk\]

Divide both by \(4hk\): \[\frac{6h + 14k - 7}{4hk}\].

The expression is now simplified. Always check for any remaining common factors to simplify further.