Juan is rewriting the expression \(12d - 26c\) as a product. Which statements about the expression are accurate and relevant to his rewriting the expression? Select two options.

A. The GCF of the numbers in each term in the expression is 2.
B. The GCF of the numbers in each term in the expression is 4.
C. The GCF of the variables in each term in the expression is \(cd\).
D. The factored expression is \(2(6d - 13c)\).
E. The factored expression is [tex]\(2 \operatorname{cod}(6 - 13)\)[/tex].



Answer :

Sure, let's analyze the expression \(12d - 26c\) to determine which statements about rewriting it as a product are accurate and relevant.

### Identifying the Greatest Common Factor (GCF)
First, we observe the coefficients of the terms:
- The coefficient of \(d\) is 12.
- The coefficient of \(c\) is 26.

To find the GCF of 12 and 26, consider the factors of each number:
- Factors of 12: \(1, 2, 3, 4, 6, 12\)
- Factors of 26: \(1, 2, 13, 26\)

The greatest common factor (GCF) is the largest number that appears in both lists of factors. Here, the GCF of 12 and 26 is 2.

### Factoring the Expression
Given the GCF is 2, we can factor the expression \(12d - 26c\) as follows:
[tex]\[ 12d - 26c = 2(6d - 13c) \][/tex]

### Let's Analyze the Options
1. The GCF of the numbers in each term in the expression is 2.
- This statement is accurate. We identified that the GCF of 12 and 26 is indeed 2.

2. The GCF of the numbers in each term in the expression is 4.
- This statement is not accurate. We determined the GCF to be 2, not 4.

3. The GCF of the variables in each term in the expression is \(cd\).
- This statement is not relevant since the variables \(d\) and \(c\) do not have common factors (they are for different terms).

4. The factored expression is \(2(6d - 13c)\).
- This statement is accurate. The expression \(12d - 26c\) can be factored into \(2(6d - 13c)\).

5. The factored expression is \(2 \operatorname{cod}(6 - 13)\).
- This statement does not make sense in the context of factoring algebraic expressions and contains elements that are not algebraically correct.

### Conclusion
The accurate and relevant statements about rewriting \(12d - 26c\) as a product are:
- The GCF of the numbers in each term in the expression is 2.
- The factored expression is [tex]\(2(6d - 13c)\)[/tex].