Question 1023732: How to factorize X^4 + 1/X^4 – 3 by completing the square. I am getting (X^2+ 1/X^2)^2 - 5. However book's answer is (x^2+1/x^2)^2 +1)(x^2+1/x^2)^2 -1)
Thank you
waqar
Found 2 solutions by mathmate, Alan3354:
Answer by mathmate(429)   (Show Source):
You can put this solution on YOUR website!
Question:
Find how many positive integers with exactly four decimal digits, that is, positive integers between 1000 and 9999 inclusive, have the following properties:
(a) are divisible by 5 or by 7 (inclusive or).
(b) are divisible by 5.
(c) are divisible by 7.
(d) are not divisible by either 5 or 7.
Solution:
We will need the inclusive/exclusive principles.
For example, we look for numbers divisible by 2 or 3 between 1 to 20.
Using integer arithmetic, i.e. discard fractions from quotients, we know that there are 20/3=6 divisible by 3, and 20/2=10 divisible by 2.
The quantity o f numbers divisible by 2 OR by 3 is NOT 6+10=16!
Why? It's because 6 is divisible by both two and three, and its multiples have been counted twice. So we must subtract 20/6=3 from 16 to get the right answer, namely 13, or 6+10-3=13, using the inclusion/exclusion principle.
Now for the pool of numbers between 1000 and 9999, we calculate the following:
(a) are divisible by 5 or by 7 (inclusive or).
Divisible by 5: 9999/5-1000/5=1999-200=1799
Divisible by 7: 9999/7-1000/7=1428-142=1286
Divisible by 35: 9999/35-1000/35=285-28=257
So
Divisible by 5 OR 7 = 1799+1286-257=2828
(b) are divisible by 5.
See part (a)
(c) are divisible by 7.
See part (a)
(d) are not divisible by either 5 or 7.
The pool of numbers is from 1000 to 9999, namely 9000 numbers, out of which 2828 are divisible by either 5, or 7 or both. The difference of 9000 and 2828 would therefore be those that are divisible neither by 5 nor 7.
Answer by Alan3354(69443)  (Show Source):
You can put this solution on YOUR website! X^4 + 1/X^4 – 3 by completing the square.
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check the book's answer:
(x^2+1/x^2)^2 +1)(x^2+1/x^2)^2 -1) = (x^2 + 1/x^2)^2 - 1
= x^4 + 2 + 1/x^4 - 1
= x^4 + 1/x^4 + 1 ---> book is wrong.
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