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f(x)= a(0) + a1(x-x(0)) +...

f(x)= a(0) + a1(x-x(0)) + a2(x-x(1))(x-x(0))I am having a hard time understanding the intuition of (x-x(1))(x-x(0)) being multiplied by the coefficient a(2). For example, if I added a(3) to the equation, I would have had to multiply a(3) by (x-x(0))(x-x(1))(x-x(2)). I've researched the Mean Value Theorem and can't make the connection. I don't understand how the multiplications keep increasing. I understand that it needs 4 points to be able to compute the 3rd divided difference. I can't seem to grasp how it works. I am a visual learner, but can slowly reason through arithmetic. Thanks for you help!

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