Question 34

The expression (6x^3 + 7x² + 1)/x is equivalent to 6x² + 7x + 1/x.

Dividing each term in the numerator (6x^3 + 7x² + 1) by x yields 6x² for the first term, 7x for the second, and 1/x for the constant term, resulting in the expression 6x² + 7x + 1/x.

A) 6^3 + 7^2 + 1/x

This option is incorrect as it incorrectly simplifies the coefficients of the polynomial terms. The expression should retain the variable x in the terms derived from the original polynomial.

B) 6^3 + 7^2 + 1

This option is also incorrect because it simplifies the coefficients to constants rather than maintaining the variable x in the polynomial terms. The correct expression must include the terms multiplied by x.

C) 6x² + 7x + 1/x

This option is correct. It accurately represents the result of dividing each term of the original polynomial by x, yielding 6x² from 6x^3, 7x from 7x², and 1/x from the constant term 1.

D) 6x² + 7x + 1

This option is incorrect as it omits the 1/x term that arises from dividing the constant term by x. The expression must include all terms to be equivalent to the original expression.

E) 6x^2 + 7x^2 + 1

This option is incorrect because it incorrectly combines the coefficients of the x² terms. The correct simplification must keep the original coefficients intact without combining them.

Conclusion

The correct answer, C) 6x² + 7x + 1/x, appropriately reflects the division of each term in the expression (6x^3 + 7x² + 1) by x. All other options fail to accurately represent the division or misinterpret the coefficients, illustrating a misunderstanding of polynomial simplification.