Factor the expression using the GCF: 15x – 25
The Correct Answer and Explanation is:
To factor the expression ( 15x – 25 ) using the Greatest Common Factor (GCF), we first need to identify the GCF of the coefficients and the variables present in the terms.
- Identify the coefficients and terms:
The expression ( 15x – 25 ) consists of two terms:
- The first term is ( 15x ).
- The second term is ( -25 ).
- Find the GCF of the coefficients:
- The coefficient of the first term, ( 15 ), can be factored as ( 3 \times 5 ).
- The coefficient of the second term, ( -25 ), can be factored as ( -1 \times 5 \times 5 ) or ( -1 \times 5^2 ). Now, let’s find the GCF:
- The prime factors of ( 15 ) are ( 3 ) and ( 5 ).
- The prime factors of ( -25 ) are ( -1 ) and ( 5^2 ). The common factor between ( 15 ) and ( 25 ) is ( 5 ). Therefore, the GCF of the coefficients ( 15 ) and ( -25 ) is ( 5 ).
- Factor out the GCF:
Next, we factor ( 5 ) out of each term in the expression:
[
15x – 25 = 5(3x) – 5(5) = 5(3x – 5)
] - Conclusion:
The factored form of the expression ( 15x – 25 ) is ( 5(3x – 5) ).
Factoring out the GCF simplifies expressions and makes it easier to work with them in algebraic manipulations. In this case, the expression can be interpreted as the product of ( 5 ) and a binomial ( (3x – 5) ). This technique not only streamlines calculations but is also foundational in solving equations, graphing functions, and understanding polynomial behavior. Recognizing the GCF is a critical skill in algebra, helping to facilitate various operations such as solving equations and simplifying expressions.