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A polynomial P has non-negative integer coefficients. If P(1)=8 and P(10)=2312 what is P(2)
Answers
Given : A polynomial P has non-negative integer coefficients.
P(1)=8 and P(10)=2312
To Find : P(2)
Solution:
P(1)=8
P(10)=2312
Let say polynomial is linear
P(x) = ax + b
P(1) = a + b = 8
P(10) = 10a + b = 2312
=> 9a = 2304
=> a= 256
=> b = -248
P(x) = 256x - 248
=> P(2) = 256(2) - 248
=> P(2) = 264
Taking polynomial Quadratic
P(x) = ax² + bx + c
P(1) = a + b + c = 8
P(10) = 100a + 10b + c = 2312
=> 99a + 9b = 2304
=> 11a + b = 256
a & b are non negative integers
P(2) = 4a + 2b + c
a = 23 , b = 3 , => c = -18 => P(2) = 80
a = 22 , b = 14 => c = -28 => P(2) = 88
a = 1 , b = 245=> c = -238 => P(2) = 256
a = 0 , b = 256 => c = -248 ( its same as linear ) => P(2) = 264
P(2) can be
80 , 88 , 96 , .................................256 , 264
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