Real-number triples turn up in the world of quadratic polynomials, where one might not expect them. To see this, let P(x) = a x2 + b x + c be a polynomial. It is completely determined by the coefficients a, b and c, which therefore form a triple in three-dimensional space. Adding two quadratic polynomials corresponds to adding their triples. Similarly, multiplying a polynomial P by a scalar corresponds to multiplying the triple representing P by that scalar. In other words, the space of quadratic polynomials and ordinary three-dimensional space are isomorphic. We can therefore associate the basis = {(1, 0, 0), (0, 1, 0), (0, 0, 1)} of three-dimensional space with a basis {I, J, K} of the space of quadratic polynomials: I is the polynomial that maps x to x2, that is, I(x) = x2; similarly, J(x) = x and K(x) = 1. This allows us to write formally P = a I + b J + c K, analogous to the expression
Although this decomposition of P cannot be given the same geometric meaning, it retains its algebraic meaning, and the diagram remains valid. These observations may seem like little more than an idle curiosity. In fact, they allow us to transfer properties of ordinary three-dimensional space to the space of quadratic polynomials. That is the beauty of abstraction! We can reason about polynomials as though they were vectors.
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Changing bases for quadratic polynomials
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