= Solution
The degree-$n$ <Bernstein basis> on $[0,1]$ is
$$
\boxed{B_i^n(t)=\binom ni t^i(1-t)^{n-i},\qquad i=0,\ldots,n.}
$$
The associated <Bézier curve> is $C(t)=\sum_iP_iB_i^n(t)$. The basis is nonnegative and has <partition of unity>, since $\sum_iB_i^n(t)=(t+1-t)^n=1$. Thus the curve is a <convex combination> of its <control points>, giving the <convex hull> enclosure useful for geometric bounds. The same identity makes the construction invariant under application of <affine functions>.
It interpolates its endpoint controls, $C(0)=P_0$, $C(1)=P_n$, and its endpoint <tangent vectors> are $n(P_1-P_0)$ and $n(P_n-P_{n-1})$. These follow from
$$
C'(t)=n\sum_{i=0}^{n-1}(P_{i+1}-P_i)B_i^{n-1}(t).
$$
The <control polygon> therefore gives direct geometric editing and tangent control. Repeated interpolation by the <De Casteljau algorithm> evaluates and subdivides the curve without converting to a monomial representation. Together, positive geometric weights, simple endpoint behavior and stable subdivision make this basis useful in <computer aided geometric design>.
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