Factor by grouping [tex]$7 m+m n+21+3 n$[/tex]
Answer (2)
The expression 7 m + mn + 21 + 3 n can be factored by grouping into the form ( m + 3 ) ( 7 + n ) . This process involves grouping terms, factoring out common factors, and then factoring out the common binomial. The final result is ( m + 3 ) ( n + 7 ) .
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Group the terms: ( 7 m + mn ) + ( 21 + 3 n ) .
Factor out common factors from each group: m ( 7 + n ) + 3 ( 7 + n ) .
Factor out the common binomial factor: ( m + 3 ) ( 7 + n ) .
The factored form is: ( m + 3 ) ( n + 7 ) .
Explanation
Understanding the Problem We are asked to factor the expression 7 m + mn + 21 + 3 n by grouping. Factoring by grouping involves rearranging terms and finding common factors to simplify the expression.
Grouping Terms First, let's group the terms: ( 7 m + mn ) + ( 21 + 3 n ) .
Factoring Each Group Now, we factor out the common factor from each group. From the first group ( 7 m + mn ) , we can factor out m , which gives us m ( 7 + n ) . From the second group ( 21 + 3 n ) , we can factor out 3 , which gives us 3 ( 7 + n ) . So, we have m ( 7 + n ) + 3 ( 7 + n ) .
Factoring out the Common Binomial Notice that ( 7 + n ) is a common binomial factor in both terms. We can factor out ( 7 + n ) from the entire expression: ( m + 3 ) ( 7 + n ) .
Final Factored Form Therefore, the factored form of the expression 7 m + mn + 21 + 3 n is ( m + 3 ) ( n + 7 ) .
Examples
Factoring by grouping is a useful technique in algebra that helps simplify complex expressions. For example, suppose you are designing a rectangular garden and want to express its area in factored form. If the area is given by 7 m + mn + 21 + 3 n , where m and n represent some dimensions, factoring this expression into ( m + 3 ) ( n + 7 ) can help you understand the possible dimensions of the garden. If m and n must be integers, you can easily find integer values for the dimensions of the garden by considering the factors of the area.
