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Cubic relationship (Posted on 2024-04-28)

Suppose two cubic polynomials f(x) and g(x) satisfy the following: f(2)=g(4), f(4)=g(8), f(8)=g(16), f(16)=g(32)+64. Find the value of g(128)-f(64).

No Solution Yet Submitted by Danish Ahmed Khan

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Solution

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Let h(x) = g(2x)-f(x)

Then h(2)=0, h(4)=0, h(8)=0, h(16)=64. And we are tasked with finding h(64).

From the three zeros h(x) = A*(x-2)*(x-4)*(x-8). Then evaluating at x=16 yields 64=A*14*12*8. Then A=1/21.

h(x) = (1/21)*(x-2)*(x-4)*(x-8). Then h(64) = (1/21)*(62)*(60)*(56) = 9920.

Posted by Brian Smith on 2024-04-28 18:45:01

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