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Let R ( x ) be the remainder when x 2 0 1 0 − x 2 0 0 9 + ( x + 2 ) 2 is divided by x 2 − 1 . Find R ( 1 ) .
This problem is a part of this set .
The answer is 9.
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5 solutions
What about the case when x is equal to - 1
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R ( -1 ) = 3
It said in the equation to solve for R ( 1 )
I have a doubt. When we simply put x=1 in the polynomial we are reduced to find the remainder when 9 is divided by 0. How is it then possible that the remainder is 9 because division by zero is not defined.
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This is using the Euclidean division for polynomials over a field, not for integers. The only assumption for the Euclidean Division for polynomials is x 2 − 1 is a nonzero polynomial. We should see the division as "divided by x 2 − 1 ", and not "divided by 0".
Let the given expression be f ( x )
According to remainder theorem
Then
f ( x ) = ( x^2 - 1 ) g (x ) + R ( x )
f ( 1 ) = 0 + R ( 1 ) ........ then
R ( 1 ) = 9
Also
f ( -1 ) = o + R ( -1 ) ...... then
R ( -1 ) = 3
Use the polynomial remainder theorem:
x 2 0 1 0 − x 2 0 0 9 + ( x + 2 ) 2 = ( x 2 − 1 ) D ( x ) + R ( x ) where D ( x ) is the dividend and R ( x ) is the remainder.
Substitute 1 into the remainder theorem and long equation on the left simplifies to 9 .
So the remainder, R ( x ) , is 9
Why we neglect x=-1 and we should let y=x^2 . If I am wrong plz correct me!
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Remainder Theorem states that P ( x ) = D ( x ) G ( x ) + R ( x )
where
P ( x ) is the polynomial to be divided, called the dividend
D ( x ) is the divisor
G ( x ) is the quotient
& R ( x ) is the remainder
For the problem, we have
P ( x ) = x 2 0 1 0 − x 2 0 0 9 + ( x + 2 ) 2
& D ( x ) = x 2 − 1
Since we are looking for R ( 1 ) , we substitute x = 1 to the equation
x 2 0 1 0 − x 2 0 0 9 + ( x + 2 ) 2 = ( x 2 − 1 ) G ( x ) + R ( x )
and we get
R ( 1 ) = 9
Just put R(x) = 1. it will give 9.
(x^2010 - x^2009 + x^2 + 4x + 4)/(x^2 - 1) where using basic algebraic method [x^2009(x - 1) + (x + 2)(x + 2)]/[(x + 1)(x -1)] so {x^2009 + [(x + 2)(x + 2)]/(x - 1)}/(x + 1) and [(x^2009)/(x + 1)] + {[(x + 2)(x + 2)]/[(x - 1)(x + 1)]} or [(x^2009)/(x + 1)] + {[(x + 2)(x + 2)]/[x^2 - 1]} since x^2010 - x^2009 = x^2009(x - 1) where if x is represented by 1 so 3^2 = 9 where 9/(1 - 1) = 9/0 where the remainder is 9 (note that the quotient is undefined or infinite) it is (clearly stated)!
You should learn how to use latex!
x 2 0 1 0 − x 2 0 0 9 + ( x + 2 ) 2 = ( x 2 − 1 ) P ( x ) + R ( x )
Substitute x = 1 and get R ( 1 ) = 9 .