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This polynomial is so extra

Source: 2017 AMC 12B #23

February 16, 2017
algebrapolynomialAMCAMC 122017 AMC 12B

Problem Statement

The graph of y=f(x)y=f(x), where f(x)f(x) is a polynomial of degree 33, contains points A(2,4)A(2,4), B(3,9)B(3,9), and C(4,16)C(4,16). Lines ABAB, ACAC, and BCBC intersect the graph again at points DD, EE, and FF, respectively, and the sum of the xx-coordinates of DD, EE, and FF is 2424. What is f(0)f(0)?
<spanclass=latexbold>(A)</span>2<spanclass=latexbold>(B)</span>0<spanclass=latexbold>(C)</span>2<spanclass=latexbold>(D)</span>245<spanclass=latexbold>(E)</span>8<span class='latex-bold'>(A) </span> -2 \qquad <span class='latex-bold'>(B) </span> 0 \qquad <span class='latex-bold'>(C) </span> 2 \qquad <span class='latex-bold'>(D) </span> \frac{24}{5} \qquad <span class='latex-bold'>(E) </span> 8