ab−ac+bd−cd−b+c
To simplify the expression \( ab - ac + bd - cd - b + c \), we can look for common factors and rearrange terms. Let's group the terms for clarity:
\[ ab - ac + bd - cd - b + c \]
Rearranging gives:
\[ (ab - ac) + (bd - cd) + (-b + c) \]
Now we can factor out common factors from each group:
From \( ab - ac \), we can factor out \( a \): \[ a(b - c) \]
From \( bd - cd \), we can factor out \( d \): \[ d(b - c) \]
The last part is \( -b + c \), which we can also rearrange to \( c - b \), or factor it out as \( -(b - c) \): \[ -(b - c) \]
Now substituting back, we have:
\[ a(b - c) + d(b - c) - (b - c) \]
We can notice that \( b - c \) is common in all terms, so we can factor that out:
\[ (b - c)(a + d - 1) \]
Thus, the simplified form of the expression \( ab - ac + bd - cd - b + c \) is:
\[ (b - c)(a + d - 1) \]
