For which value of u will (u² − 81)x¹³ + (u − 9)x⁶ + 5 have degree 0?

Correct answer: u = 9

A polynomial has degree 0 when it reduces to a nonzero constant. Therefore both positive-power terms must vanish: u² − 81 = 0 for the x^13 term and u − 9 = 0 for the x^6 term. The second condition immediately gives u = 9. Checking it in the first condition, 9² − 81 = 81 − 81 = 0, so both variable terms disappear and the expression becomes the nonzero constant 5. Its degree is therefore 0, making option A correct. Although u = −9 satisfies u² − 81 = 0, it gives u − 9 = −18, so a nonzero x^6 term remains and the degree is 6. Values 0 and 81 satisfy neither required pair of conditions.