Perform the following additions and/or subtractions:
$\left(3 w^3-4 w-1\right)-\left(-3 w^3+4 w+1\right)=$
Answer (1)
Distribute the negative sign: ( 3 w 3 − 4 w − 1 ) − ( − 3 w 3 + 4 w + 1 ) = 3 w 3 − 4 w − 1 + 3 w 3 − 4 w − 1 .
Combine the w 3 terms: 3 w 3 + 3 w 3 = 6 w 3 .
Combine the w terms: − 4 w − 4 w = − 8 w .
Combine the constant terms: − 1 − 1 = − 2 . The final expression is 6 w 3 − 8 w − 2 .
Explanation
Understanding the problem We are asked to simplify the expression ( 3 w 3 − 4 w − 1 ) − ( − 3 w 3 + 4 w + 1 ) . This involves distributing the negative sign and combining like terms.
Distributing the negative sign First, distribute the negative sign in the second term: ( 3 w 3 − 4 w − 1 ) − ( − 3 w 3 + 4 w + 1 ) = 3 w 3 − 4 w − 1 + 3 w 3 − 4 w − 1
Combining like terms Next, combine the like terms: ( 3 w 3 + 3 w 3 ) + ( − 4 w − 4 w ) + ( − 1 − 1 ) = 6 w 3 − 8 w − 2
Final result Therefore, the simplified expression is 6 w 3 − 8 w − 2 .
Examples
Polynomials are used to model curves and shapes in various fields such as engineering, computer graphics, and physics. For example, engineers use polynomials to design bridges and buildings, while computer graphics designers use them to create realistic images. Understanding how to simplify polynomial expressions is crucial in these applications.
