Factor and simplify to obtain a product-of-sums expression with four term

Factor and simplify to obtain a product-of-sums expression with four terms: abc′ + d′e + ace + b′c′d′

The correct answer and explanation is :

To simplify the given Boolean expression and express it in a product-of-sums (POS) form with four terms, we follow these steps:

Given Boolean Expression:

[
abc’ + d’e + ace + b’c’d’
]

Step 1: Identify Common Factors

We analyze the terms to group and factor common variables:

1. Grouping terms strategically:
   [
   (abc’ + ace) + (d’e + b’c’d’)
   ]
2. Factor within each group:

• From (abc’ + ace), factor out (a):
  [
  a(bc’ + ce)
  ]
• From (d’e + b’c’d’), factor out (d’):
  [
  d'(e + b’c’)
  ]

Step 2: Further Factorization

Looking at the term ( bc’ + ce ), we try factoring in a way that leads to a sum of products:

1. Rewriting ( bc’ + ce ):
   [
   bc’ + ce = c'(b) + c(e) = c'(b) + c(e)
   ]
2. Rewriting ( e + b’c’ ) (which is already in sum form).

Thus, our new factored expression is:

[
(a(c’ + e)(b + c)) (d’ + (e + b’c’))
]

Final Expression in Product-of-Sums Form:

[
(a + d’)(c’ + e)(b + c)(e + b’c’)
]

---

Explanation:

1. Factoring is crucial: We carefully group terms to extract common factors, simplifying step by step.
2. Converting to POS form: The goal is to express the equation as a product of sum terms, ensuring it follows the desired format.
3. Logical equivalence: The transformation preserves the original logic but presents it in a structured, minimal format.

Thus, the correct Product-of-Sums expression is:

[
(a + d’)(c’ + e)(b + c)(e + b’c’)
]