The polynomial $f (x) = x^{4} - 2 x^{3} + 3 x^{2} - a x + b$ when divided by $(x - 1)$ and $(x + 1)$ leaves the remainders $5$ and $19$ respectively . Find the value of $a$ and $b$ . Hence , find the remainder when $f (x)$ is divided by $(x - 3)$ .

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Solution

A polynomial f(x) is divided by a linear polynomial (x-a) leaves remainder which equal f(a).
so, $f (x) = x^{4} - 2 x^{3} + 3 x^{2} - a x + b$
so, when it is divided by (x-1) it will leave a remainder
$f (1) = 5 \Rightarrow a - b = - 3 ⟶ (1) s i m i l a r l y, f (- 1) = 19 \Rightarrow a + b = 13 ⟶ (2) s o l v i n g (1) a n d (2) w e g e t a = 5 a n d b = 8 ∴ t h e p o l y n o m i a l i s f (x) = x^{4} - 2 x^{3} + 3 x^{2} - 5 x + 8 a l s o w h e n i t i s d i v i d e d b y (x - 3) l e a v e s r e m a i n d e r f (3) = 47$

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