Simplify: [tex]$\frac{6 x^5-2 x-4}{2 x}$[/tex]
Answer (2)
The simplified expression for 2 x 6 x 5 − 2 x − 4 is 3 x 4 − 1 − x 2 . This result is obtained by dividing each term of the numerator by the denominator and simplifying the fractions. Each component of the expression is calculated step by step to arrive at the final simplified form.
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Divide each term in the numerator by 2 x .
Simplify the expression: 2 x 6 x 5 − 2 x 2 x − 2 x 4 = 3 x 4 − 1 − x 2 .
Compare the simplified expression with the given options.
The simplified expression is 3 x 4 − 1 − x 2 .
Explanation
Understanding the Problem We are asked to simplify the expression 2 x 6 x 5 − 2 x − 4 . This involves dividing each term in the numerator by the denominator, 2 x .
Dividing Each Term We divide each term in the numerator by 2 x :
2 x 6 x 5 − 2 x 2 x − 2 x 4 Simplifying each fraction: 2 6 ⋅ x x 5 − 2 2 ⋅ x x − 2 4 ⋅ x 1 3 x 5 − 1 − 1 − x 2 3 x 4 − 1 − x 2
Finding the Correct Option Comparing the simplified expression 3 x 4 − 1 − x 2 with the given options, we find that it matches the fourth option.
Final Answer Therefore, the simplified expression is 3 x 4 − 1 − x 2 .
Examples
Simplifying rational expressions is a fundamental skill in algebra. For instance, when designing a bridge, engineers use rational expressions to model the load distribution. Simplifying these expressions allows them to efficiently calculate stress and ensure the bridge's structural integrity. Similarly, in economics, simplifying rational expressions can help in analyzing cost functions and optimizing production processes.
