Simplify: $-7 z^4-2 z+2+2 z^4+5 z-5$
Answer (1)
Combine the z 4 terms: − 7 z 4 + 2 z 4 = − 5 z 4 .
Combine the z terms: − 2 z + 5 z = 3 z .
Combine the constant terms: 2 − 5 = − 3 .
The simplified expression is − 5 z 4 + 3 z − 3 .
Explanation
Understanding the Problem We are asked to simplify the expression − 7 z 4 − 2 z + 2 + 2 z 4 + 5 z − 5 . This involves combining like terms, which are terms with the same variable raised to the same power.
Grouping Like Terms First, let's group the like terms together: ( − 7 z 4 + 2 z 4 ) + ( − 2 z + 5 z ) + ( 2 − 5 ) .
Combining z 4 Terms Now, combine the z 4 terms: − 7 z 4 + 2 z 4 = ( − 7 + 2 ) z 4 = − 5 z 4 .
Combining z Terms Next, combine the z terms: − 2 z + 5 z = ( − 2 + 5 ) z = 3 z .
Combining Constant Terms Finally, combine the constant terms: 2 − 5 = − 3 .
Final Simplified Expression Putting it all together, the simplified expression is: − 5 z 4 + 3 z − 3 .
Examples
Simplifying polynomial expressions is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use polynomials to model the behavior of structures and systems. Simplifying these expressions allows them to analyze and design more efficient and reliable systems. Consider a scenario where an engineer is designing a bridge and needs to calculate the total load on a support beam. The load can be represented by a polynomial expression, and simplifying this expression helps the engineer determine the maximum stress on the beam, ensuring the bridge's safety and stability.
