Perform the indicated operations.
$\left[\left(9 m^2+3 m-6\right)-\left(7 m^2-5 m+7\right)\right]-\left(m^2+m+5\right)$
Answer (2)
Distribute the negative sign in the first set of parentheses.
Combine like terms within the first set of parentheses.
Substitute the simplified expression back into the original expression.
Distribute the negative sign in the second set of parentheses and combine like terms to get the final simplified expression: m 2 + 7 m − 18 .
Explanation
Understanding the Problem We are asked to perform the indicated operations and simplify the given polynomial expression: [ ( 9 m 2 + 3 m − 6 ) − ( 7 m 2 − 5 m + 7 ) ] − ( m 2 + m + 5 )
Simplifying the Inner Parentheses First, we simplify the expression inside the square brackets. We distribute the negative sign in the first set of parentheses: ( 9 m 2 + 3 m − 6 ) − ( 7 m 2 − 5 m + 7 ) = 9 m 2 + 3 m − 6 − 7 m 2 + 5 m − 7
Combining Like Terms Next, we combine like terms within the parentheses: 9 m 2 − 7 m 2 + 3 m + 5 m − 6 − 7 = 2 m 2 + 8 m − 13
Substituting Back Now, we substitute the simplified expression back into the original expression: [ 2 m 2 + 8 m − 13 ] − ( m 2 + m + 5 )
Distributing the Negative Sign We distribute the negative sign in the second set of parentheses: 2 m 2 + 8 m − 13 − m 2 − m − 5
Combining Like Terms Finally, we combine like terms: 2 m 2 − m 2 + 8 m − m − 13 − 5 = m 2 + 7 m − 18
Examples
Polynomials are used to model various real-world phenomena, such as the trajectory of a ball, the design of roller coasters, and even economic trends. Simplifying polynomial expressions allows engineers and scientists to make accurate predictions and design efficient systems. For example, if m represents the number of items sold, the simplified expression m 2 + 7 m − 18 could represent the profit made, and understanding this expression helps in making business decisions.
The given expression simplifies to m 2 + 7 m − 18 . This was achieved by distributing negative signs, combining like terms, and simplifying step-by-step. The final result is boxed for clarity.
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