Which of the following expressions is equivalent to (6x^3 + 7x² + 1)/x?

Rationale

(6x^3 + 7x² + 1)/x = 6x² + 7x + 1/x.

Dividing each term in the polynomial \(6x^3 + 7x² + 1\) by \(x\) simplifies the expression to \(6x^2 + 7x + \frac{1}{x}\), as each term's degree is reduced by one.

A (6^3 + 7^2 + 1/x) YOUR ANSWER

This expression incorrectly raises the coefficients to their respective powers instead of maintaining their linear forms in relation to \(x\). The correct simplification should involve dividing the entire polynomial by \(x\), not altering the coefficients in this manner.

B (6^3 + 7^2 + 1) YOUR ANSWER

Similar to choice A, this option incorrectly simplifies the coefficients, treating them as constants rather than variables dependent on \(x\). The correct expression must include the variable \(x\) in the terms derived from the original polynomial.

C (6x² + 7x + 1/x) CORRECT

This choice accurately represents the simplified form of the original polynomial after dividing each term by \(x\). Each term is correctly reduced, preserving the mathematical relationship established in the original expression.

D (6x² + 7x + 1) YOUR ANSWER

While this expression correctly simplifies the first two terms, it omits the \(\frac{1}{x}\) term, which is essential after dividing the constant term \(1\) by \(x\). Thus, it fails to represent the complete simplification of the original expression.

E (6x^2 + 7x^2 + 1) YOUR ANSWER

This expression incorrectly combines the \(x^2\) terms, leading to a misrepresentation of the original polynomial. The correct simplification should keep the distinct coefficients of \(x^2\) separate rather than summing them incorrectly.

Conclusion

The expression \((6x^3 + 7x² + 1)/x\) simplifies directly to \(6x² + 7x + \frac{1}{x}\). Correctly reducing each term by \(x\) yields the only option that accurately reflects this simplification, which is choice C. The other options either misinterpret the coefficients or omit important terms, demonstrating the need for careful algebraic manipulation.