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Question 1204811: Fill in the blanks with positive integers:
(3 + sqrt(5))*3*(5 + 3*sqrt(5))^3 = ___ + ___* sqrt(5)
Found 3 solutions by MathLover1, ikleyn, MathTherapy:
Answer by MathLover1(20855)   (Show Source):
Answer by ikleyn(53937)  (Show Source):
You can put this solution on YOUR website! .
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.
To make writing easier, replace  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  = 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  = 5.
I get then "second integer number in the answer" = 648*5 + 2400 = which is easy to compute = 5640.
Therefore, the final ANSWER is  (**)
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.
Solved.
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For safety of your mind, simply ignore the post by @MathLover1,
since it is WRONG.
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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.
Answer by MathTherapy(10858)  (Show Source):
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