How Do You Evaluate the Product of g(x) at the Roots of a Quintic Polynomial?

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In summary, evaluating g(r1)g(r2)g(r3)g(r4)g(r5) allows for analysis of particle interactions in a system to gain insight into its structure and properties. g(r) is calculated by comparing the number of particle pairs at a given distance to the expected number in a random system. It can determine the density of the system and provide information about its structure based on the shape of the g(r) curve. However, limitations include its focus on pairwise interactions and potential influences from system size and external factors.
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anemone
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Here is this week's POTW:

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Let $r_1,\,r_2,\,\cdots,r_5$ be the roots of $f(x)=x^5+x^2+1$ and $g(x)=x^2-2$. Evaluate $g(r_1)g(r_2)g(r_3)g(r_4)g(r_5)$.

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Congratulations to lfdahl for his correct solution (Cool) !
\[\coprod_{i=1}^{5}g(r_i) = \coprod_{i=1}^{5}(r_i^2-2) = \coprod_{i=1}^{5}(r_i-\sqrt{2})(r_i+\sqrt{2})=\coprod_{i=1}^{5}(r_i-\sqrt{2})\coprod_{i=1}^{5}(r_i+\sqrt{2})\\\\ = \left ( (-1)^5\coprod_{i=1}^{5}(\sqrt{2}-r_i) \right )\left ( (-1)^5\coprod_{i=1}^{5}(-\sqrt{2}-r_i) \right ) = f(\sqrt{2})f(-\sqrt{2}) =\left ( \left ( \sqrt{2} \right )^5+3 \right )\left ( -\left ( \sqrt{2} \right )^5+3 \right ) \\\\= -2^5+9 = -32 + 9 = -23.\]
 

Related to How Do You Evaluate the Product of g(x) at the Roots of a Quintic Polynomial?

1. What is the purpose of evaluating g(r1)g(r2)g(r3)g(r4)g(r5)?

The purpose of evaluating g(r1)g(r2)g(r3)g(r4)g(r5) is to determine the spatial correlation between five particles in a system. This helps in understanding the behavior and properties of the system.

2. How is g(r) related to the radial distribution function?

g(r) is the normalized form of the radial distribution function, which describes the probability of finding a particle at a certain distance from another particle in a system. g(r) is calculated by dividing the radial distribution function by the average number of particles in a shell at that distance.

3. What information can be obtained from the values of g(r1)g(r2)g(r3)g(r4)g(r5)?

The values of g(r1)g(r2)g(r3)g(r4)g(r5) provide information about the spatial arrangement of the particles in the system. It can also indicate the presence of any ordering or clustering of particles at specific distances.

4. How is g(r) calculated in a simulation or experiment?

In a simulation, g(r) is calculated by counting the number of particles in a spherical shell at a certain distance from a reference particle and then normalizing it by the average number of particles in that shell. In an experiment, g(r) can be obtained by analyzing the diffraction pattern of the system using techniques such as X-ray or neutron scattering.

5. Can g(r1)g(r2)g(r3)g(r4)g(r5) be used to determine the type of interactions between particles?

Yes, the values of g(r1)g(r2)g(r3)g(r4)g(r5) can provide information about the type of interactions between particles. For example, a sharp peak at a specific distance indicates strong attractive interactions, while a dip indicates repulsive interactions. However, g(r) alone is not enough to determine the exact nature of interactions and other factors should also be considered.

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