When f(x) is divided by (x−1)(x+2), the remainder is (x+3), and when f(x) is divided by (x^2 +2x+5), the remainder is (2x+1). Find the remainder when f(x) is divided by (x−1)(x^2 +2x+5).


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Question Number 120091 by ZiYangLee last updated on 29/Oct/20

$When f (x) is divided by (x - 1) (x + 2),$ $the remainder is (x + 3), and when f (x)$ $is divided by (x^{2} + 2 x + 5), the remainder$ $is (2 x + 1) .$ $Find the remainder when f (x) is divided$ $by (x - 1) (x^{2} + 2 x + 5) .$
	Answered by TITA last updated on 29/Oct/20

	$for a given polynomial p (x) = f (x) q (x) + r$ $w h e r e r i s t h e r e m a i n d e r$ $if r = o then q (x) divids p (x)$ $let f (x) = p (x) (x - 1) (x + 2) + x + 3 . . . . . . . (1)$ $f (x) = p (x) (x^{2} + 2 x + 5) + 2 x + 1 . . . . . . . . (2)$ $(1) - (2) \Rightarrow 0 = p (x) [x^{2} + x - 2 - (x^{2} + 2 x + 5)] + 2 - x$ $\Rightarrow x - 2 = p (x) [- x - 7]$ $e n c e p (x) = \frac{2 - x}{x + 7}$ $s i n c e f (x) = p (x) q (x) + r$ $h e n c e r = f (x) - p (x) q (x)$ $(2) \Rightarrow f (x) = (\frac{2 - x}{x + 7}) (x^{2} + 2 x + 5) + 2 x + 1$ $\Rightarrow r = [(\frac{2 - x}{x + 7}) (x^{2} + 2 x + 5) + 2 x + 1] - [(\frac{2 - x}{x + 7}) (x - 1) (x^{2} + 2 x + 5)]$ $\Rightarrow r = (\frac{2 - x}{x + 7}) (x^{2} + 2 x + 5) [2 - x] + 2 x + 1$ $\Rightarrow r = \frac{{(2 - x)}^{2} (x^{2} + 2 x + 5)}{x + 7} + 2 x + 1$
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