Which expression can be used to find the difference of the polynomials?
[tex](10 m-6)-(7 m-4)[/tex]
A. [tex][10 m+(-7 m)]+[(-6)+4][/tex]
B. [tex](10 m+7 m)+[(-6)+(-4)][/tex]
C. [tex][(-10 m)+(-7 m)]+(6+4)[/tex]
D. [tex][10 m+(-7 m)]+[6+(-4)][/tex]
Answer (1)
Distribute the negative sign: ( 10 m − 6 ) − ( 7 m − 4 ) = 10 m − 6 − 7 m + 4 .
Group like terms: 10 m − 7 m − 6 + 4 .
Rewrite using addition: 10 m + ( − 7 m ) + ( − 6 ) + 4 .
Group terms with m and constants: [ 10 m + ( − 7 m )] + [( − 6 ) + 4 ] .
The correct expression is [ 10 m + ( − 7 m )] + [( − 6 ) + 4 ] .
Explanation
Understanding the Problem We are given the expression ( 10 m − 6 ) − ( 7 m − 4 ) and four options. We want to find the expression that is equivalent to the given expression.
Plan of Action To find the difference of the polynomials, we need to distribute the negative sign and combine like terms.
Distributing the Negative Sign First, distribute the negative sign: ( 10 m − 6 ) − ( 7 m − 4 ) = 10 m − 6 − 7 m + 4
Grouping Like Terms Next, group like terms: 10 m − 7 m − 6 + 4
Rewriting with Addition Rewrite the expression using addition: 10 m + ( − 7 m ) + ( − 6 ) + 4
Grouping Terms Group the terms with m and the constant terms: [ 10 m + ( − 7 m )] + [( − 6 ) + 4 ]
Comparing with Options Now, we compare our resulting expression, [ 10 m + ( − 7 m )] + [( − 6 ) + 4 ] , with the given options. The first option, [ 10 m + ( − 7 m )] + [( − 6 ) + 4 ] , matches our result.
Final Answer Therefore, the expression that can be used to find the difference of the polynomials is: [ 10 m + ( − 7 m )] + [( − 6 ) + 4 ]
Examples
Polynomial subtraction is used in various real-world scenarios, such as calculating profit margins in business. For example, if a company's revenue is represented by the polynomial 10 m − 6 and its costs are represented by the polynomial 7 m − 4 , the profit can be found by subtracting the cost polynomial from the revenue polynomial, resulting in ( 10 m − 6 ) − ( 7 m − 4 ) . This simplifies to 3 m − 2 , where m could represent the number of items sold.
