The factorization of [tex]x y+y+6 x+6[/tex] is:
A. [tex](x+y)(x+1)[/tex]
B. [tex](x+y)(y+6)[/tex]
C. [tex](x+6)(y+1)[/tex]
D. [tex](x+1)(y+6)[/tex]
Answer (1)
To factor the expression x y + y + 6 x + 6 , we use factoring by grouping:
Group the terms: ( x y + y ) + ( 6 x + 6 ) .
Factor out common factors from each group: y ( x + 1 ) + 6 ( x + 1 ) .
Factor out the common binomial factor ( x + 1 ) : ( x + 1 ) ( y + 6 ) .
The factorization is ( x + 1 ) ( y + 6 ) .
Explanation
Understanding the Problem We are asked to factor the expression x y + y + 6 x + 6 . Factoring is the process of breaking down an expression into a product of simpler expressions. In this case, we will use factoring by grouping.
Grouping Terms First, we group the terms in pairs: ( x y + y ) + ( 6 x + 6 ) .
Factoring out Common Factors Next, we factor out the greatest common factor (GCF) from each group. From the first group ( x y + y ) , the GCF is y , so we have y ( x + 1 ) . From the second group ( 6 x + 6 ) , the GCF is 6 , so we have 6 ( x + 1 ) . Thus, the expression becomes y ( x + 1 ) + 6 ( x + 1 ) .
Factoring out the Common Binomial Now, we notice that ( x + 1 ) is a common factor in both terms. We factor out ( x + 1 ) from the entire expression: ( x + 1 ) ( y + 6 ) .
Final Factorization Therefore, the factorization of x y + y + 6 x + 6 is ( x + 1 ) ( y + 6 ) .
Examples
Factoring expressions like x y + y + 6 x + 6 is useful in various real-world scenarios. For example, suppose you are designing a rectangular garden where the area can be expressed as x y + y + 6 x + 6 . By factoring this expression into ( x + 1 ) ( y + 6 ) , you can determine the possible dimensions of the garden. If you know the area and one of the dimensions, you can easily find the other dimension. This is also applicable in business when analyzing costs and revenues, where factoring can help simplify complex expressions to understand the relationships between different variables.
