Bezier curve written as a polynomial: each point is weighted; the curve is the sum of weighted points. The weights vary with t. For example, at t=0, the curve is at A, and the contribution of points B, C, D... is zero. A influences the curve less and less as t increases.
"Show All Point Weights" displays a vector in the direction of each point; the length of each vector show how far the curve is pushed in that direction.
A tool for giving a nth order Bézier from n+1 control points: http://www.geogebratube.org/material/show/id/83845

At higher orders the curve becomes less flexible. Nearby points average each other out, and the range of influence of each point is very small. most of the time, most weights are near zero. We have the greatest control over the endpoints.
There are three elements to the equation of the curve.

Binomial coefficients Ck
For a curve of order n, there are n+1 points.
Multiply the points by the values of Pascal's Triangle: the (binomial) coefficients of .

Control points Pk

Powers of t and (1-t), Tk

Example:
If n=3, the binomial expansion is (1+x)³ = 1 + 3x + 3x² +x³.
1. Coefficients Ck = {1, 3, 3, 1}
2. Pk = {A, B, C, D}
3. Let τ = 1-t. Then Ct = {τ³, τ² t, τ t², t³}
Multiply corresponding elements: all the first elements, all the second elements.... and add.
FlanParty = Zip[c*P*t, c, Ck, P, Pk, t, Tk]
= {1*A*τ³, 3*B*(τ² t) , 3*C*(τ t²), 1*D*t³}.
Add 'em:
Sum[FlanParty] = (Aτ³) + (3B t τ²) + (3C t² τ) + (D t³)
Identify the locus:
Locus[S, t] is the Bezier curve of order n, with the given control points A-D.
It is the locus of all points which satisfy the polynomial constructed in the first two steps.
...but the curves don't work very well. So why don't we simplify the whole business? (Next)
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Bézier Curves
1. Construction: http://www.geogebratube.org/material/show/id/27320→2. Higher Order Curves
3. Basic Implementation: http://www.geogebratube.org/material/show/id/27631
4. Weights: http://www.geogebratube.org/material/show/id/28313
5. Continuous path: http://www.geogebratube.org/material/show/id/81193
6. ...