Question 7

6x³ + 7x² + 1/x is equivalent to itself.

The expression 6x³ + 7x² + 1/x remains unchanged when compared to the other options, confirming that it is indeed equivalent to itself.

A) 63 + 72 + 1/x

This expression modifies the coefficients of the polynomial terms incorrectly. It incorrectly states 63 and 72 instead of 6 and 7, making it an incorrect equivalence to the original expression.

B) 63 + 72 + 1

Similar to Option A, this expression also changes the coefficients of the polynomial terms to 63 and 72, and it does not include the variable term 1/x at all, thus failing to represent the original expression accurately.

C) 6x³ + 7x² + 1/x

This option is the same as the original expression and therefore is indeed equivalent. It accurately represents the same polynomial and rational term without any alteration.

D) 6x² + 7x + 1

This option alters the degrees of the polynomial terms, reducing the degrees of x from 3 and 2 to 2 and 1, respectively. Additionally, it omits the rational term 1/x, rendering it a different expression altogether.

E) 6x² + 7x² + 1

This expression mistakenly combines the x² terms, resulting in 13x² instead of keeping them separate. It also lacks the 1/x term, making it an incorrect representation of the original expression.

Conclusion

Option C stands out as the only expression that accurately reflects the original expression, 6x³ + 7x² + 1/x, without any alterations. All other options misrepresent the original in terms of coefficients, degrees, or omission of terms, confirming that they are not equivalent.