Find infinitely many polynomials f(x) in Z3[x] such that f(a)=0 for all a in Z3
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The polynomial x (x - 1) (x - 2) = x (x^2 - 3x + 2) = x^3 - 3x^2 + 2x certainly has this property.
And since -3 = 0 in Z_3, this is the polynomial x^3 + 2x. OK, that's just one example.
But any polynomial of the form q(x) (x^3 + 2x) will *also* have this property. If you plug in x = 0, 1, or 2, you will get 0 in the second factor and hence 0 no matter what q(0), q(1), or q(2) is. And there are certainly infinitely many nonzero polynomials q(x) to choose from in Z_3[x].
So
{q(x) (x^3 + 2x): q[x] in Z_3[x], and q(x) is nonzero}
is an infinite set of polynomials in Z_3[x] that take the value 0 for all a in Z_3. If you want a smaller and more explicit set of polynomials, for example, the polynomials
x^n (x^3 + 2x), n = 1, 2, 3, ...
all have this property.
And since -3 = 0 in Z_3, this is the polynomial x^3 + 2x. OK, that's just one example.
But any polynomial of the form q(x) (x^3 + 2x) will *also* have this property. If you plug in x = 0, 1, or 2, you will get 0 in the second factor and hence 0 no matter what q(0), q(1), or q(2) is. And there are certainly infinitely many nonzero polynomials q(x) to choose from in Z_3[x].
So
{q(x) (x^3 + 2x): q[x] in Z_3[x], and q(x) is nonzero}
is an infinite set of polynomials in Z_3[x] that take the value 0 for all a in Z_3. If you want a smaller and more explicit set of polynomials, for example, the polynomials
x^n (x^3 + 2x), n = 1, 2, 3, ...
all have this property.