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Answer to Question #11033 in Real Analysis for sravya

Question #11033
polynomial f(x)= x4-2x3+3x2-ax+b when divided by (x-1) and (x+1) leaves reminder 5 and 19.Find the values of A and B and the reminder when f(x) is divided by (x-2)
Expert's answer
we use theorem f(x) divided by x-c leaves reminder which equal to f(a)
so
f(1)=5, f(-1)=19
1^4-2*1^3+3*1^2-a*1+b=5
and
(-1)^4-2*(-1)^3+3*(-1)^2-a*(-1)+b=19
2-a+b=5 and 6+a+b=19 so a=b-3
from first equation
6+b-3+b=19
b=8,
a=5,
f(x)=x^4-2x^3+3x^2-5x+8
and reminder when f(x) is divided by x-2 is
f(2)=10

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