Factor by grouping.
$7 z^2+21 z-a z-3 a$
Answer (2)
Group the terms: ( 7 z 2 + 21 z ) + ( − a z − 3 a ) .
Factor out the GCF from each group: 7 z ( z + 3 ) − a ( z + 3 ) .
Factor out the common binomial factor: ( 7 z − a ) ( z + 3 ) .
The factored expression is ( 7 z − a ) ( z + 3 ) .
Explanation
Understanding the Problem We are asked to factor the expression 7 z 2 + 21 z − a z − 3 a by grouping. This means we want to rearrange and group terms in such a way that we can factor out a common binomial factor.
Grouping Terms First, we group the terms: ( 7 z 2 + 21 z ) + ( − a z − 3 a ) .
Factoring out GCF Next, we factor out the greatest common factor (GCF) from each group. From the first group, 7 z 2 + 21 z , the GCF is 7 z . Factoring this out, we get 7 z ( z + 3 ) . From the second group, − a z − 3 a , the GCF is − a . Factoring this out, we get − a ( z + 3 ) . So the expression becomes 7 z ( z + 3 ) − a ( z + 3 ) .
Factoring out the Common Binomial Now, we notice that ( z + 3 ) is a common binomial factor in both terms. We factor out ( z + 3 ) from the entire expression: ( 7 z − a ) ( z + 3 ) .
Final Answer Therefore, the factored expression is ( 7 z − a ) ( z + 3 ) .
Examples
Factoring by grouping is a useful technique in algebra that helps simplify complex expressions. For instance, imagine you're designing a rectangular garden where the area is represented by the expression 7 z 2 + 21 z − a z − 3 a . By factoring this expression into ( 7 z − a ) ( z + 3 ) , you determine that one side of the garden has a length of ( 7 z − a ) and the other side has a length of ( z + 3 ) . This allows you to plan the dimensions of your garden efficiently based on the variables involved.
To factor the expression 7 z 2 + 21 z − a z − 3 a by grouping, we group the terms and factor out the GCF from each part. The result is the expression factored as ( 7 z − a ) ( z + 3 ) .
;
