To factor the expression [tex]\(28 + 56t + 28w\)[/tex], let's proceed step-by-step.
1. Identify the Greatest Common Factor (GCF)
- We first notice that each term in the expression [tex]\(28 + 56t + 28w\)[/tex] has a common factor. Specifically, the GCF of 28, 56, and 28 is 28.
2. Factor out the GCF
- We can rewrite each term as a multiple of 28:
[tex]\(28 = 28 \cdot 1\)[/tex]
[tex]\(56t = 28 \cdot 2t\)[/tex]
[tex]\(28w = 28 \cdot w\)[/tex]
- Thus, we can factor 28 out of the expression:
[tex]\(28 + 56t + 28w = 28 (1 + 2t + w)\)[/tex]
3. Review the choices offered
- A: [tex]\(2(56 + 112t + 56w)\)[/tex]
- B: [tex]\(28(1 + 2t + w)\)[/tex]
- C: [tex]\(4(7 + 13t + 7w)\)[/tex]
- D: [tex]\(7(4 + 8t + 4w)\)[/tex]
4. Verify the Factored Forms
- Choice A: [tex]\(2(56 + 112t + 56w)\)[/tex]
Simplifying this, we get [tex]\(2 \cdot 56 + 2 \cdot 112t + 2 \cdot 56w = 112 + 224t + 112w\)[/tex]. This is not equivalent to [tex]\(28 + 56t + 28w\)[/tex].
- Choice B: [tex]\(28(1 + 2t + w)\)[/tex]
Expanding this, we get [tex]\(28 \cdot 1 + 28 \cdot 2t + 28 \cdot w = 28 + 56t + 28w\)[/tex]. This is exactly the original expression.
- Choice C: [tex]\(4(7 + 13t + 7w)\)[/tex]
Simplifying this, we get [tex]\(4 \cdot 7 + 4 \cdot 13t + 4 \cdot 7w = 28 + 52t + 28w\)[/tex]. This is not equivalent to [tex]\(28 + 56t + 28w\)[/tex].
- Choice D: [tex]\(7(4 + 8t + 4w)\)[/tex]
Expanding this, we get [tex]\(7 \cdot 4 + 7 \cdot 8t + 7 \cdot 4w = 28 + 56t + 28w\)[/tex]. This is exactly the original expression.
Thus, the two expressions that are equivalent to [tex]\(28 + 56t + 28w\)[/tex] are:
- B: [tex]\(28(1 + 2t + w)\)[/tex]
- D: [tex]\(7(4 + 8t + 4w)\)[/tex]
