Which of the following expressions is equivalent to: 6x³ + 7x² + 1/x?

Rationale

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

This expression is already in its simplest form and accurately represents the polynomial and rational function it denotes. Each term retains its original coefficients and variables, ensuring that no alterations have been made to the expression itself.

A (63 + 72 + 1/x) YOUR ANSWER

This choice presents incorrect coefficients for the polynomial terms. The terms 63 and 72 do not correspond to the original expression's coefficients of 6 and 7, respectively. Therefore, this expression does not maintain the equivalence of the original.

B (63 + 72 + 1) YOUR ANSWER

This option disregards the variable components entirely, reducing the expression to a constant value. The original expression contains variable terms, which are essential for maintaining the equivalence, rendering this choice invalid.

C (6x³ + 7x² + 1/x) CORRECT

This choice directly matches the original expression, retaining all coefficients and variables. It accurately reflects the polynomial and rational nature of the initial expression without any alterations, thus confirming its equivalence.

D (6x² + 7x + 1) YOUR ANSWER

This option misrepresents the original expression by altering both the powers of x and the coefficients. The terms x² and the constant 1 are not accurate representations of the original expression, which includes x³ and 1/x.

E (6x² + 7x² + 1) YOUR ANSWER

This choice incorrectly combines terms; while it maintains the correct coefficients for x², it fails to reflect the original expression's x³ term and the 1/x component. The result is a significant alteration that invalidates equivalence.

Conclusion

In conclusion, the expression 6x³ + 7x² + 1/x is equivalent to itself, as it accurately represents its terms without modification. All other options either misrepresent the coefficients and variable powers or reduce the expression to constants, failing to capture the original's true form. Thus, understanding expressions' equivalence is critical in algebraic manipulations and problem-solving.