If (p(x)=(a^2-9)x^4+2x+1) is a linear polynomial, which values of (a) are possible?

Correct answer: (a=3) or (a=-3)

The direct answer is C: \\(a=3\\) or \\(a=-3\\). A linear polynomial has degree 1, so the coefficient of every power higher than 1 must be zero, while the coefficient of x must be nonzero. Here \\(p(x)=(a^2-9)x^4+2x+1\\). The coefficient of \\(x^4\\) is \\(a^2-9\\). Set it equal to zero: \\(a^2-9=0\\). Hence \\(a^2=9\\), so \\(a=3\\) or \\(a=-3\\). For either value, the expression becomes \\(0x^4+2x+1=2x+1\\), which has degree 1 because the coefficient of x is 2, not zero. Therefore option C is correct. Option A, \\(a=0\\), gives coefficient \\(-9\\), leaving an \\(x^4\\) term, so the polynomial is degree 4. Option B, \\(a=9\\), gives coefficient 72, again leaving the fourth-degree term. Option D is wrong because two valid values have been found. Do not set \\(a\\) itself to zero; set the coefficient of the unwanted highest-power term to zero. Memory cue: for a polynomial to become linear, remove all powers 2 and above, then confirm the x coefficient remains nonzero.