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Division Algorithm for a Polynomial

when the poly...

Question

when the polynomial x^4 + 2x^3 + 8x^2 + 12x + 18 is divided by another polynomial x^2 + 5 , if the remainder comes out to be px - q , find the values of p and q.

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Solution

Dear student,

$x^{2} + 5) x^{4} + 2 x^{3} + 8 x^{2} + 12 x + 18 (x^{2} + 2 x + 3 x^{4} + 5 x^{2} \bar{2 x^{3} + 3 x^{2} + 12 x + 18} 2 x^{3} + 10 x \bar{3 x^{2} + 2 x + 18} 3 x^{2} + 15 \bar{2 x + 3} T h e r e m a i n d e r i n t h e a b o v e d i v i s i o n i s 2 x + 3, t h e g i v e n r e m a i n d e r i s p x - q ∴ p x - q = 2 x + 3 p x + (- q) = 2 x + 3 p = 3, - q = 3, q = - 3$

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