Fill in the blanks with positive integers:
(3 + sqrt(5))*3*(5 + 3*sqrt(5))^3 = ___ + ___* sqrt(5)
To make writing easier, replace √5
by x in the left side of the original expression. After replacing, you will get this polynomial (3+x)*3*(5+3x))^3.
(3 + x)*3*(5 + 3x))^3 = 81x^4 + 648x^3 + 1890x^2 + 2400x + 1125. (*)
Now, to get first integer number of the right side, I consider the terms of this polynomial (*) with even degrees
even(x) = 81x^4 + 1890x^2 + 1125
and substitute there x2 = 5.
I get then "second integer number in the answer" = 648*5 + 2400 = which is easy to compute = 5640. Therefore, the final ANSWER is 12600 + 5640* √5 (**)
Finally, to check the answer, I calculated (**) and compared with the left side of the original expression. I got the same number, which confirms correctness of my solution.