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Algebra Difficulty 4.3 AIME Find the answer United States

Problem:

Let P(x)P(x) be a polynomial of degree at most 33 such that P(x)=11+x+x2P(x) = \frac{1}{1 + x + x^{2}} for x=1,2,3,4x = 1, 2, 3, 4. What is P(5)P(5)?

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:

Answer: −391\frac{-3}{91}

The forward difference of a polynomial PP is ΔP(x)=P(x+1)−P(x)\Delta P(x) = P(x+1) - P(x), which is a new polynomial with degree reduced by one. Therefore, if we apply this operation three times we'll get a constant function, and we can work back up to get a value of P(5)P(5). Practically, we create the following table of differences:
1317113121−421−691−82733427310273−24273 \begin{array}{ccccccc} \frac{1}{3} & & \frac{1}{7} & & \frac{1}{13} & & \frac{1}{21} \\ & \frac{-4}{21} & & \frac{-6}{91} & & \frac{-8}{273} & \\ & \frac{34}{273} & & \frac{10}{273} & & \\ & & \frac{-24}{273} & & & \end{array}
Then extend it to be the following table:

Figure 1

So our answer is −9273=−391\frac{-9}{273} = \frac{-3}{91}

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.