Certainly! Let's start by understanding the given expression and then simplify and factor it step-by-step.
Given expression:
[tex]\[ a(b - c) + d(c - b) \][/tex]
1. Identify the structure:
We notice that the expression consists of two terms:
- The first term is [tex]\( a(b - c) \)[/tex]
- The second term is [tex]\( d(c - b) \)[/tex]
2. Rewriting the second term:
Notice that [tex]\( d(c - b) \)[/tex] can be rewritten as [tex]\( d(-1)(b - c) \)[/tex], which simplifies to [tex]\( -d(b - c) \)[/tex].
Thus, the expression becomes:
[tex]\[ a(b - c) + (-d)(b - c) \][/tex]
[tex]\[ a(b - c) - d(b - c) \][/tex]
3. Factor out the common term:
We see that [tex]\( (b - c) \)[/tex] is a common factor in both terms. So, we can factor [tex]\( (b - c) \)[/tex] out from the expression:
Factor out [tex]\( (b - c) \)[/tex]:
[tex]\[ (b - c)(a - d) \][/tex]
Therefore, the expression [tex]\( a(b - c) + d(c - b) \)[/tex] can be written as the product of two polynomials:
[tex]\[ (a - d)(b - c) \][/tex]
