Consider the polynomials f(t) = t + 1 and g(t) = (t + 2)(t + k), where k is an arbitrary constant. For which values of the constant k are the three polynomials f(t), tf(t) and g(t) a basis of P2?
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This will happen when the matrix of coefficients is nonsingular. Evaluate its determinant, and find the values of for which it is nonzero.
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