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I am trying to solve the equation $$ 3894937908247641871050398074967894254 = 764008325721660_x$$

Here is my attempt $$ 7x^{14} + 6x^{13} + 4x^{12} + 0x^{11} + 0x^{10} + 8x^9 + 3x^8 + 2x^7 + 5x^6 + 7x^5 + 2x^4 + 1x^3 + 6x^2 + 6x + 0 - 3894937908247641871050398074967894254 = 0$$

Wolfram Alpha gave $x = 357.412$ but when I solve for $x$ I get $6.75367976499\times 10^{31}$

Can someone please tell me what I'm missing?

user193661
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1 Answers1

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The answer $x=357.412$ is only an approximation to six significant digits. That is, all you know from what Alpha told you is that $357.4115 \leq x < 357.4125$.

So if you just look at the leading term of your polynomial, the exact value could be anywhere between $7 \times 357.4115^{14}$ and $7 \times 357.4125^{14}$. The difference between these two numbers has magnitude greater than $1.5 \times 10^{32}$, so you should not be surprised if you find an error of about $6.75 \times 10^{31}$ in your calculation.

And that is even without considering whether the calculation of the polynomial is done with $37$ significant digits, which is what you would need in order to have an answer accurate to the nearest integer.

David K
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  • So an initially small difference becomes giant when _. What is this effect called? Exponential growth? – user193661 Jul 15 '15 at 01:52
  • It's mostly the exponential growth of $357.412^n$ that makes the difference so large in the end. The numbers $357.4115$ and $357.4125$ differ by only a small fraction of a percent, and even after raising to the 14th power the percentage difference is still small, but it's a small percentage of a really big number $(7 \times 357.412^{14} > 3 \times 10^{36}),$ and the "bigness" of that number completely overwhelms the "smallness" of the percentage. – David K Jul 15 '15 at 02:34