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find remainder x power 100is divided by xpower2-3x+2

find remainder x power 100is divided by xpower2-3x+2


 


 


 


 


 


 


 


 


 


 

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1 Answers

Latika Leekha
askIITians Faculty 165 Points
9 years ago
Let us assume a(x) = x100 and b(x) = x2-3x+2
We need to find the remainder P(x) when a(x) is divided by b(x).
a(x) = b(x)Q(x) + P(x), where 0 ≤ deg P(x) < deg b(x)
Now
x100 = (x2-3x+2) Q(x) + P(x)
= (x-1)(x-2) Q(x) + (ax+b) (Since 0< deg P(x) <2, so P(x) is of the form ax + b)
x100 = (x-1)(x-2) Q(x) + (ax+b)
Now put x = 1, we get
a + b = 1
Putting x = 2, we get
2a + b = 2100
On solving these two equations, we obtain the values of a and b as
a = 2100 - 1
b = 2- 2100
Hence, the required reminader is
P(x) = ax+b = (2100 – 1)x + (2- 2100)
Thansk & Regards
Latika Leekha
askIITians Faculty

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