# Approximating polynomial roots with summations?

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I have been working with my friend in economics (always fun ) and we are trying to find polynomial roots using summations.

The reason I ask is because we are modelling an economy as a system of equations, and if there is a surplus product then it becomes a polynomial.

For example, we have an economy of two sectors (wheat and coal). We have the relationship

280 qr. Wheat + 12 t. coal --> 575 qr. Wheat

120 qr. Wheat + 8 t. coal --> 20 t. coal

We then set up the value per unit Wheat as X and value per unit coal as Y, giving us:

[Math]\frac{(1+r)(280X + 12Y)}{(1+r)(120X+8Y)} = \frac{575X}{20Y}[/Math]

thus by reduction of the rate of profit ("1+r"), we multiply both sides out to receive

[Math]20Y(280X + 12Y) = 575X(120X+8Y)[/Math]

and by setting X=1 (because we assume f(Y)=X to get the value of coal in terms of wheat) we get a quadratic expression. In this case, we can just plug this into the quadratic equation and sha-zam, done.

Yet what if we calculate out the value of the units wheat and coal in terms of the dated inputs? Intuitively, this would make no difference (from the economist's perspective).

Yet mathematically, how would one portray this? Wouldn't it merely be an approximation of polynomial roots with a summation?

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You can just use either eqtn to solve for one of the variables, then substitute that into the second equation and get an answer for one of them. After that you would just use that answer in either eqtn to get the value for the other variable.

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