What should be added in the polynomial $x^{3} - 6 x^{2} + 11 x + 8$ so that it is completely divisible by $x^{2} - 3 x + 2$ ?

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Solution

$x^{2} - 3 x + 2 = (x - 1) (x - 2) h e n c e g i v e n p o l y n o m i a l m u s t b e d i v i s i b l e b y b o t h x = 1 a n d 2 (w h i c h a r e i t s z e r o) p u t x = 1 1 - 6 + 11 + 8 = 14 p u t x = 2 2^{3} - 6 (2^{2}) + 11 \times 2 + 8 = 8 - 24 + 22 + 8 = 14 H e n c e 14 i s r e m a i n d e r a n d i t m u s t b e s u b t r a c t e d f r o m g i v e n p o l y n o m i a l$

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Find teh number which should be added to $x^{2} + x + 3$ so that the resulting polynomial is completely divisible by (x+3).

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