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Remainder Theorem

Find the poly...

Question

Find the polynomial
A polynomial of degree 3 vanishes at x - 1 and x = 3. It leaves remainder - 2 & - 1, when divided by (x + 1), (x - 2) resp. Find polynomial

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Solution

$S i n c e t h e p o l y n o m i a l v a n i s h e s o r b e c o m e 0 a t x = 1 a n d x = 3, t h e r e f o r e, (x - 1) a n d (x - 3) a r e i t s f a c t o r s . L e t t h e o t h e r f a c t o r b e (x - α) L e t p (x) = k (x - α) (x - 1) (x - 3) U \sin g r e m a i n d e r t h e o r e m, i f p (x) i s d i v i d e d b y (x + 1) i . e . (x - (- 1)), t h e n r e m a i n d e r i s g i v e n b y p (- 1) p (- 1) = k (- 1 - α) (- 1 - 1) (- 1 - 3) = - 2 8 k (1 + α) = 2(1) A g a i n U \sin g r e m a i n d e r t h e o r e m, i f p (x) i s d i v i d e d b y (x - 2), t h e n r e m a i n d e r i s g i v e n b y p (2) p (2) = k (2 - α) (2 - 1) (2 - 3) = - 2 k (2 - α) = 2_(2) F r o m e q u a t i o n 1 a n d 2 k (2 - α) = 8 k (1 + α) (2 - α) = 8 (1 + α) (2 - α) = 8 + 8 α 9 α = - 6 α = - \frac{2}{3} P u t i n e q u a t i o n 1 8 k (1 - \frac{2}{3}) = 2 8 k (\frac{1}{3}) = 2 k = \frac{3}{4} p (x) = \frac{3}{4} (x + \frac{2}{3}) (x - 1) (x - 3)$

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