To factor the expression \( 48y + 36z \) using the greatest common factor (GCF), we first find the GCF of the coefficients (48 and 36).
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Finding the GCF:
- The prime factorization of 48 is \( 2^4 \times 3^1 \).
- The prime factorization of 36 is \( 2^2 \times 3^2 \).
- The GCF is taken by taking the lowest power of each prime:
- For \( 2 \): minimum of \( 4 \) (from 48) and \( 2 \) (from 36) is \( 2^2 \).
- For \( 3 \): minimum of \( 1 \) (from 48) and \( 2 \) (from 36) is \( 3^1 \).
- Thus, \( \text{GCF} = 2^2 \times 3^1 = 4 \times 3 = 12 \).
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Factoring out the GCF:
- Now we factor \( 12 \) out of \( 48y + 36z \): \[ 48y + 36z = 12(4y) + 12(3z) = 12(4y + 3z) \]
Thus, the expression \( 48y + 36z \) can be factored as \( 12(4y + 3z) \).
None of the options provided appear exactly as \( 12(4y + 3z) \), but we can also observe:
- \( 4(12y + 9z) \) expands back to \( 48y + 36z \).
- Thus, the factorization could also be \( 12(3y + 2z) \), which is simply another form.
The correct answer would be: 12(4y + 3z) (if this representation is listed) follows from the calculations above and represents the factorization correctly based on \( 12 \) as a GCF.
If the options specify differing forms that ultimately yield the same relation, assess accordingly in the context of factors and GCF.