if x=3+2√2 then prove (x⁶+x⁴+x²+1)/x³
Answers
Step-by-step explanation:
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शत्रुबुद्धिविनाशाय दीपकाय नमोऽस्तु ते।।
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Answer:
Yes. It will always be positive.
When you factorise;
x^8-x^6+x^4-x^2+1
= x^6(x^2-1)+x^2(x^2-1) + 1
=(x^2-1)(x^6+x^2)+1
For Real values of ‘x’
if x ≥ 1 or x ≤ -1
(x^6+x^2) is always positive for both negative and positive values (negative value raised to even power gives a positive value).
(x^2–1) is also positive or equal to 0.
So the whole expression (x^2-1)(x^6+x^2)+1 is also positive
If -1 < x < 1 ,
(x^2–1) is a negative fraction greater than 0 less than 1. ( as x^2<1 for x^2 is a positive fraction betwewn 0 and 1)
(x^6+x^2) is a positive fraction greater than 0, less than 1.(as x^2 and x^6 are positive fractions between 0 and 1)
So the product (x^2-1)(x^6+x^2) is a negative fraction within the interval (0,1) {as 2 fractions within intervals (0,1) when multiplied gives back a fraction within limits (0,1) }
So the whole expression (x^2-1)(x^6+x^2)+1 is positive.[as,(x^2-1)(x^6+x^2)< 1]
Thus, for every real value of ‘x’ the given expression is positive.
For Imaginary (Complex) values of ‘x’,
x^2, x^6 would be negative.
{As square root on a negative value(i.e a Complex number) raised to an even power always yields negative value.
PROOF: Let √-a be a Complex number. Now,
√-a = i a where i = √-1, a > 0
(√-a)^(2n) =(a)^(2n)×( i )^(2n)
[where ‘2n’ is an even number]
=(a)^(2n)×-1= -(a)^(2n) = a negative number
[ (a)^(2n) is positive as ‘a’ is positive] }
So, both expressions (x^2-1) and (x^6+x^2) will be negative value resulting in a positive product and thus giving the whole value of (x^2-1)(x^6+x^2)+1 as positive.
Step-by-step explanation:
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