With respect to (z), what is the degree of (9x^6z^{10}-8x^2z^7+5x^3-6)?

Correct answer: (10)

The phrase “with respect to \(z\)” tells us which variable is being treated as the polynomial variable. The degree is the greatest exponent of \(z\) having a non-zero coefficient. Expressions involving other variables, such as \(x^6\), are treated as coefficients when the degree is considered with respect to \(z\).

In \(9x^6z^{10}-8x^2z^7+5x^3-6\), the powers of \(z\) are 10, 7, 0, and 0. The coefficient of \(z^{10}\) is \(9x^6\), which is non-zero as a polynomial coefficient, so the highest relevant exponent is 10. Therefore option C is correct. The exponent 6 belongs to \(x\), not \(z\), and the constant terms have degree zero in \(z\).