Find the value of α and β , if the division of x2010+αx2011+β by (x−1) leaves the remainder zero and by (x+1) leaves the remainder 4.
Given polynomial p(x) = x2010 + αx2011 + β
Now, as the remainder is zero when p(x) is divided by (x- 1). So put x = 1.
(1)2010 + α(1)2011 + β = 0
1 + α + β = 0
α + β = -1.... (1)
Similarly, put x = -1, In this case the remainder is 4
(-1)2010 + α(-1)2011 + β = 4
1 - α + β = 4
β - α = 3... (2)
Adding (1) and (2) we get β = 1 and α = -2