Question 1204811
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Fill in the blanks with positive integers:
(3 + sqrt(5))*3*(5 + 3*sqrt(5))^3 = ___ + ___* sqrt(5)
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        What is written in the post by @MathLover1, is  FATALLY  INCORRECT 

        starting from the third line of her post to the last line,  including.


        I came to bring a correct solution.



<pre>
To make writing easier, replace {{{sqrt(5)}}} by x in the left side of the original expression.

After replacing, you will get this polynomial  (3+x)*3*(5+3x))^3.


It is

    (3 + x)*3*(5 + 3x))^3 = 81x^4 + 648x^3 + 1890x^2 + 2400x + 1125.    (*)


You can do multiplications manually - it is a mechanical procedure.

I myself, caring about my mind, did it using an online calculator
https://www.emathhelp.net/en/calculators/algebra-1/multiplying-polynomials-calculator/


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  {{{x^2}}} = 5.  

I get then  "first integer number in the answer" = 81*5^2 + 1890*5 + 1125 = which is easy to compute = 12600.


Next, to get second integer number of the right side, I consider the terms of this polynomial (*)
with odd degrees

    odd(x) = 648x^3 + 2400x

and substitute there  {{{x}}} = 5.  

I get then  "second integer number in the answer" = 648*5 + 2400 = which is easy to compute = 5640.


Therefore, the final <U>ANSWER</U> is  {{{12600 + 5640*sqrt(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.
</pre>

Solved.


---------------


For safety of your mind, simply ignore the post by @MathLover1,
since it is WRONG.



\\\\\\\\\\\\\\\\\\



What is shown in my post is HOW TO organize your work and HOW TO do it in other similar cases.


Multiplying polynomials of high degree by hand is a torture, which will add 
nothing to your mind and/or to your skills, and here is the way how to avoid it.


Now, in the XXI century, people/students do it differently than they did it in the past, in the XX century and before.