Questions: Completely factor the expression by grouping, if possible. 3ac-5bd+bc-15ad

Transcript text: Completely factor the expression by grouping, if possible. \[ 3 a c-5 b d+b c-15 a d \]

Solution

Solution Steps

Solution Approach

To factor the expression by grouping, we first rearrange the terms to see if we can group them into pairs that have a common factor. We then factor out the greatest common factor from each pair. If the resulting expression has a common binomial factor, we can factor that out to complete the factorization.

Step 1: Rearranging the Expression

We start with the expression: \[ 3ac - 5bd + bc - 15ad \] We can rearrange the terms to group them effectively: \[ (3ac - 15ad) + (bc - 5bd) \]

Step 2: Factoring Each Group

Next, we factor out the greatest common factor from each group:

From the first group \(3ac - 15ad\), we can factor out \(3a\): \[ 3a(c - 5d) \]
From the second group \(bc - 5bd\), we can factor out \(b\): \[ b(c - 5d) \]

Step 3: Combining the Factored Groups

Now we can combine the factored groups: \[ 3a(c - 5d) + b(c - 5d) \] Since both terms contain the common factor \((c - 5d)\), we can factor that out: \[ (3a + b)(c - 5d) \]