Discussion
Let \(R_{n}[x]\) denote the vector space of all polynomials over R having degree atmost \(n\). Then the dimension of \(\mathbf{R}_{n}[x]\) is :
Answer: B
The dimension is \(n + 1\).
Every polynomial of degree at most \(n\) is uniquely \(a_{0} + a_{1}x + \cdots + a_{n}x^{n}\), so \(\{1, x, x^{2}, \ldots, x^{n}\}\) is a basis. It has \(n + 1\) elements, because the constant term is counted as well as the powers 1 to \(n\).
The answer \(n\) forgets the constant polynomial 1, \(n^{2}\) has no connection with this space, and \(\infty\) is the dimension of the space of all polynomials, with no bound on the degree.
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