Simplify $\left(-s^2+s-1\right)-\left(s^2-s+1\right)$.
Answer (1)
Distribute the negative sign: ( − s 2 + s − 1 ) − ( s 2 − s + 1 ) = − s 2 + s − 1 − s 2 + s − 1 .
Combine the s 2 terms: − s 2 − s 2 = − 2 s 2 .
Combine the s terms: s + s = 2 s .
Combine the constant terms: − 1 − 1 = − 2 . The simplified expression is − 2 s 2 + 2 s − 2 .
Explanation
Understanding the Problem We are asked to simplify the expression ( − s 2 + s − 1 ) − ( s 2 − s + 1 ) . This involves distributing the negative sign and combining like terms.
Distributing the Negative Sign First, distribute the negative sign in the second term: ( − s 2 + s − 1 ) − ( s 2 − s + 1 ) = − s 2 + s − 1 − s 2 + s − 1
Combining Like Terms Next, combine like terms. We have the s 2 terms, the s terms, and the constant terms: ( − s 2 − s 2 ) + ( s + s ) + ( − 1 − 1 ) = − 2 s 2 + 2 s − 2
Final Answer Therefore, the simplified expression is − 2 s 2 + 2 s − 2 .
Examples
Simplifying algebraic expressions is a fundamental skill in mathematics and has many practical applications. For example, if you are calculating the area of a garden and need to subtract one area from another, you would use similar simplification techniques. Suppose you have a rectangular garden with area A 1 = ( x 2 + 3 x + 2 ) and you want to remove a smaller rectangular area A 2 = ( x 2 + x + 1 ) to create a new flower bed. The area of the remaining garden would be A 1 − A 2 , which simplifies to ( x 2 + 3 x + 2 ) − ( x 2 + x + 1 ) = 2 x + 1 . This kind of algebraic manipulation is essential in various fields, including engineering, physics, and computer science.
