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If p(x)=x^4-2x^3+3x^2-ax+b is a polynomial such that when it is divided by x-1 and x+1 the remainders are respectively 5 and 19 .determine the remainder

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P(x) = x⁴ - 2 x³ + 3 x² - a x + b

reminder of the polynomial when divided by (x - k) is P(k). substitute k in place of x.

P(1) = 5 => 1 -2 +3 - a + b = 5 b - a = 3

P(-1) = 19 => 1 +2 + 3 + a + b = 19 a + b = 13 adding the two equations we get: 2 b = 16 => b = 8 a = 5

reminder when P(x) is divided by (x-2) : = P(2) = 2^4 - 2 2^3 + 3 2^2 - 5 * 2 + 8 = 10