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Any help would be appreciated

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There is a unique polynomial p(x) of degree 4 with rational coefficients and leading coefficient 1 which has sqrt(2) + sqrt(5) as a root. What is p(1)?

Aug 13, 2019

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$$\text{The coefficient of x^3 is -1 times the sum of the roots}\\ \text{The constant term is the product of all the roots}\\ \text{Thus if the coefficients are rational and \sqrt{2}+\sqrt{5} is a root then so must the following be roots}\\ -\sqrt{2}+\sqrt{5},~\sqrt{2}-\sqrt{5},~-\sqrt{2}-\sqrt{5}$$

$$\text{Using those 4 roots expand out \prod \limits_{k=1}^4 (x-r_k) and find the coefficient of the x^3 term}\\ \text{This is the negative of p(1) i.e. the sum of the coefficients of the quartic}\\ \text{You should get the sum is equal to 2\sqrt{5}}$$

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Aug 13, 2019
edited by Rom  Aug 13, 2019
edited by Rom  Aug 13, 2019