The polynomial p (x) = x^4 - 2x^3 +3x^2 - a x + b when divided by (x - 1) and (x + 1) leaves the remainder 5 and 19 respectively. Find the values of ' a ' and ' b '. Also find the remainder when p (x) is divided by (x - 2).
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A Polynomial f(x) if divided by a linear polynomial (x-a) leaves remainder which equals f(a).
So , getting back to our question -
f(x) = x^4 - 2x^3 + 3x^2 - ax + b
So , when it is divided by (x - 1) it’ll leave a remainder = f(1) = 5 (Given).
f(1) = 1^4 - 2×1^3 + 3×1^2 - a×1 + b = 5
=> 1 - 2 + 3 - a + b = 5
=> a - b = (-3) …. Eqn(1)
Now , Similarly -
f(-1) = (-1)^4 - 2×(-1)^3 + 3×(-1)^2 - a×(-1) + b = 19
=> 1 + 2 + 3 + a + b = 19
=> a + b = 13 …. Eqn(2)
Now , adding equations (1) and (2) , We’ll get -
(a+b) + (a-b) = (-3) + 13
=> 2a = 10 => a = 5
So , (a +b) = 13 implies b = 8
Hence , Values of a and b are 5 and 8 respectively.
Done!
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