Factor 12x+60 using the greatest common factor.
The Correct Answer and Explanation is:
To factor the expression ( 12x + 60 ) using the greatest common factor (GCF), we follow these steps:
Step 1: Identify the greatest common factor (GCF)
The first step in factoring any expression is to identify the GCF of all the terms in the expression.
- The two terms in the expression are ( 12x ) and ( 60 ).
- For ( 12x ), the coefficient is 12 and the variable is ( x ).
- For ( 60 ), the coefficient is 60, and there is no variable.
Now, let’s determine the GCF of the numerical coefficients (12 and 60):
- The factors of 12 are: ( 1, 2, 3, 4, 6, 12 ).
- The factors of 60 are: ( 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 ).
The common factors between 12 and 60 are ( 1, 2, 3, 4, 6, 12 ), and the greatest of these is 12.
Since the term ( 12x ) includes the variable ( x ) and ( 60 ) does not, the GCF of the entire expression is 12.
Step 2: Factor out the GCF
Next, we factor the GCF (which is 12) from both terms of the expression.
The expression ( 12x + 60 ) can be factored as follows:
[
12x + 60 = 12(x + 5)
]
Step 3: Verify the factorization
To check if the factorization is correct, distribute the ( 12 ) across the terms inside the parentheses:
[
12(x + 5) = 12x + 60
]
This matches the original expression, so the factorization is correct.
Final Answer:
The factored form of ( 12x + 60 ) is:
[
12(x + 5)
]
Explanation:
Factoring by the greatest common factor (GCF) involves identifying the largest factor that is common to each term in the expression. In this case, the GCF of 12 and 60 is 12, so we factor out 12 from the expression. This simplifies the expression to ( 12(x + 5) ), where ( x + 5 ) is the remaining factor. This method is useful for simplifying expressions and solving equations efficiently.