Which expression can be used to find the difference of the polynomials?
$(4 m-5)-(6 m-7+2 n)$
$(4 m-5)+(6 m+7+2 n)$
$(4 m-5)+(-6 m+7+2 n)$
$(4 m-5)+(-6 m-7-2 n)$
$(4 m-5)+(-6 m+7-2 n)$
Answer (1)
Distribute the negative sign: ( 4 m − 5 ) − ( 6 m − 7 + 2 n ) = ( 4 m − 5 ) + ( − 6 m + 7 − 2 n ) .
Simplify the expression: ( 4 m − 5 ) + ( − 6 m + 7 − 2 n ) .
The correct expression is: ( 4 m − 5 ) + ( − 6 m + 7 − 2 n ) .
Explanation
Understanding the Problem We are given the expression ( 4 m − 5 ) − ( 6 m − 7 + 2 n ) and asked to find an equivalent expression.
Type of Problem The problem involves subtracting one polynomial from another.
Key Strategy We need to distribute the negative sign correctly.
Subtracting Polynomials To find the difference of the polynomials ( 4 m − 5 ) and ( 6 m − 7 + 2 n ) , we need to subtract the second polynomial from the first. This means we need to distribute the negative sign to each term in the second polynomial.
Distributing the Negative Sign Distributing the negative sign, we have: ( 4 m − 5 ) − ( 6 m − 7 + 2 n ) = ( 4 m − 5 ) + ( − 6 m + 7 − 2 n ) So, the expression is ( 4 m − 5 ) + ( − 6 m + 7 − 2 n ) .
Finding the Correct Option Comparing this with the given options, we see that the correct expression is ( 4 m − 5 ) + ( − 6 m + 7 − 2 n ) .
Final Answer The expression that can be used to find the difference of the polynomials is ( 4 m − 5 ) + ( − 6 m + 7 − 2 n ) .
Examples
Polynomial subtraction is used in various real-world applications, such as calculating profit margins in business. For example, if a company's revenue is represented by the polynomial 4 m − 5 and its costs are represented by the polynomial 6 m − 7 + 2 n , then the profit can be found by subtracting the cost polynomial from the revenue polynomial, resulting in ( 4 m − 5 ) − ( 6 m − 7 + 2 n ) . This simplifies to − 2 m + 2 − 2 n , which represents the company's profit.
