IF x does not equal -1
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mohit_1607
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srcc25anu
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1-x^16 can be broken down using the formula a^2 - b^2 = (a-b)*(a+b)
therefore 1-x^16 = (1+x^8)*(1-x^8)
Now 1-x^8 can be (1+x^4)*(1-x^4)
Now 1-x^4 can be (1+x^2)*(1-x^2)
Now 1-x^2 can be (1+x)*(1-x)
So (1-x^16) = (1+x^8)*(1+x^4)*(1+x^2)*(1+x)*(1-x)
Denominator = (1+x^8)*(1+x^4)*(1+x^2)*(1+x)
Dividing only leaves (1-x)
Hence ans D
therefore 1-x^16 = (1+x^8)*(1-x^8)
Now 1-x^8 can be (1+x^4)*(1-x^4)
Now 1-x^4 can be (1+x^2)*(1-x^2)
Now 1-x^2 can be (1+x)*(1-x)
So (1-x^16) = (1+x^8)*(1+x^4)*(1+x^2)*(1+x)*(1-x)
Denominator = (1+x^8)*(1+x^4)*(1+x^2)*(1+x)
Dividing only leaves (1-x)
Hence ans D
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Plug in and ballpark.bpolley00 wrote:If X does not equal -1 then 1-x^16/(1+X)(1+X^2)(1+X^4)(1+X^8) is equal to
A) -1
B) 1
C) X
D) 1-x
E) X-1
Can someone touch on this
Let x=10.
Since the numerator and the denominator will be extremely large, we can ignore all of the 1's, which will have little impact on our approximation.
The expression becomes:
(-10¹�)/(10*10²*10�*10�) = (-10¹�)/(10¹�) = -10.
Thus, when x=10 is plugged into the correct answer choice, the result must be close in value to -10.
Only D works:
1-x = 1-10 = -9.
The correct answer is D.
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Atekihcan
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Note that if we multiply the denominator with (1 - x), it will cause a chain of multiplications and the denominator will become (1 - x¹�) as (1 - x)(1 + x) = (1 - x²) and (1- x²)(1 + x²) = (1 - x�) and so on.
So let us multiply both the numerator and denominator wit (1 - x) and we can safely do that as x ≠1.
So, the given expression can be written as (1 - x)(1 - x¹�)/(1 - x¹�) = (1 - x)
Answer : D
So let us multiply both the numerator and denominator wit (1 - x) and we can safely do that as x ≠1.
So, the given expression can be written as (1 - x)(1 - x¹�)/(1 - x¹�) = (1 - x)
Answer : D
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Manpreet Singh
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Remember basic algebra rule
(a^2 - b^2)= (a-b)*(a+b)
we are given (1-x^16)it can be written as (1-x^8)(1+x^8)
Further elaborating
(1-x^4)(1+x^4)(1+x^8)
Now
(1-x^2)(1+x^2)(1+x^4)(1+x^8)
Finally (1-x)(1+x)(1+x^2)(1+x^4)(1+x^8)
So (1-x)(1+x)(1+x^2)(1+x^4)(1+x^8)/(1+x)(1+x^2)(1+x^4)(1+x^8)
so we are left with 1-x
Ans is D
(a^2 - b^2)= (a-b)*(a+b)
we are given (1-x^16)it can be written as (1-x^8)(1+x^8)
Further elaborating
(1-x^4)(1+x^4)(1+x^8)
Now
(1-x^2)(1+x^2)(1+x^4)(1+x^8)
Finally (1-x)(1+x)(1+x^2)(1+x^4)(1+x^8)
So (1-x)(1+x)(1+x^2)(1+x^4)(1+x^8)/(1+x)(1+x^2)(1+x^4)(1+x^8)
so we are left with 1-x
Ans is D
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Here's another approach.bpolley00 wrote:If X does not equal -1 then 1-x^16/(1+X)(1+X^2)(1+X^4)(1+X^8) is equal to
A) -1
B) 1
C) X
D) 1-x
E) X-1
Can someone touch on this
Plug "nice" values into to the given expression and evaluate.
Try x = 0
1-x^16/(1+X)(1+X^2)(1+X^4)(1+X^8) = 1-0^16/(1+0)(1+0^2)(1+0^4)(1+0^8)
= 1
So, when x = 0, the expression evaluates to be 1
The correct answer will be an expression that also evaluates to be 1, when x = 0.
When we plug 0 into the 5 answer choices, only B and D evaluate to be 1
So, we can eliminate A, C and E.
Now try another "nice" value for x.
Try x = 1
1-x^16/(1+X)(1+X^2)(1+X^4)(1+X^8) = 1-1^16/(1+1)(1+1^2)(1+1^4)(1+2^8)
= 0
When we plug 1 into answer choices B and D, only D evaluates to be 0
So, the correct answer is D
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Brent
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