Find the remainder of the division of the polynomial $x^{3} + 2 x^{2} - 2 x + 5 b y x^{2} - 1$ without performing the division.

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Solution

let the dividend be R and remainder be ax+b

therefore $(x^{2} - 1) R + (a x + b) = x^{3} + 2 x^{2} - 2 x + 5$

substituting $x = 1$ and -1 we get $a = - 1, b = 7$

therefore the remainder is $- x + 7$

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