Problem 77 - Introductory Algebra for College Students

Question

Factor by grouping. $$x y+5 x+9 y+45$$

Step-by-Step Solution

Answer

The factored form of \(x y+5 x+9 y+45\) by grouping is \( (y+5)(x+9)\).

Step 1: Identify the groups

The given expression, \(x y+5 x+9 y+45\), can be grouped into two pairs: \(x y+5 x\) and \(9 y+45\). The plan is to factor out the common factors from each group.

Step 2: Factor out the common factors

The common factor in the first group \(x y+5 x\) is \(x\), so factoring out \(x\), the first group becomes \(x(y+5)\). In the second group \(9 y+45\), the common factor is 9, so it becomes 9(y+5).

Step 3: Check the factored expressions

Looking at the factored groups, both have \(y+5\) as a part of the expression. This indicates that the grouping was correct. So, continue to the next step.

Step 4: Complete the Factor by Grouping Process

Take the common binomial factor \(y+5\) out from both factored groups which gives the final factored form as, \( (y+5)(x+9)\).