SOLUTION: Find a polynomial of degree 4 with intger coefifients that has 2 square root + 5 square root as a root.

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Question 37501: Find a polynomial of degree 4 with intger coefifients that has 2 square root + 5 square root as a root.
Answer by venugopalramana(3286) About Me  (Show Source):
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Find a polynomial of degree 4 with intger coefifients that has 2 square root + 5 square root as a root.
FIRST LET US PUT A QUADRATIC WITH (2^0.5+5^0.5) AND -(2^0.5+5^0.5) AS ROOTS WE GET THE QUADRATIC AS
{X+(2^0.5+5^0.5)}{X-(2^0.5+5^0.5)}=0
=X^2-(2^0.5+5^0.5)^2=X^2-2-5-2*(10^0.5)=(X^2-7)-2*(10^0.5)=0
NOW LET US MULTIPLY WITH THE CONJUGATE AGAIN
{(X^2-7)-2*(10^0.5)}{(X^2-7)+2*(10^0.5)}=0
=(X^2-7)^2-4*10=0
X^4-14X^2+49-40=0
X^4-14X^2+9=0