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2. Suppose that p_0,...p_m are polynomials in the space P_m(F) (of polynomials over F with degree at most m) such that p_j(2)=0 for eachj. Prove that the set {p_0,...,p_m} is not lin. independant in P_m(F).

let $\displaystyle p_j(x) \in P_m(\mathbb{F}), \ j=0, \cdots, m,$ with a common root $\displaystyle \alpha.$ (in your problem $\displaystyle \alpha=2.$) then $\displaystyle q_j(x)=\frac{p_j(x)}{x-\alpha} \in P_{m-1}(\mathbb{F}), \ j=0, \cdots , m.$ but $\displaystyle \dim_{\mathbb{F}} P_{m-1}(\mathbb{F})=m,$ and