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# Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following. (i) $$p(x) = x^3 - 3x^2 + 5x - 3 , g(x)= x^2 - 2$$ (ii) $$p(x) = x^4 - 3x^2 + 4x + 5 , g(x)= x^2 - x + 1$$ (iii) $$p(x) = x^4 - 5x + 6 , g(x)= 2 - x^2$$

(i) Solving it using long division:
$$\begin{array}{rrrr|ll} x^3 & - 3x^2 & + 5x & - 3 & x^2 - 2\\ -x^3 & & + 2x & & x - 3 \\ \hline & -3x^2 & +7x & -3\\ & \phantom{-}3x^2 & & -6 & & & & \\ \hline & & +7x & -9 \\ \hline \end{array}$$
Here, Quotient = $$x - 3$$ and Remainder = $$7x - 9$$.

(ii) Solving it using long division:
$$\begin{array}{rrrr|ll} x^4 & - 3x^2 & + 4x & + 5 & x^2 - x + 1\\ x^3 -x^4 & -x^2 & & & x^2 + x - 3 \\ \hline x^3 & - 4x^2 & + 4x & + 5\\ \phantom{-}x^3 & + x^2 & -x & & & \\ \hline & -3x^2 & + 3x & +5 \\ & +3x^2 & -3x & +3 \\ \hline & & & 8 \\ \hline \end{array}$$
Here, Quotient = $$x^2 + x - 3$$ and Remainder = $$8$$.

(iii) Solving it using long division:
$$\begin{array}{rrrr|ll} &x^4 & - 5x &+ 6 &-x^2 + 2\\ &2x^2 - x^4 & & & -x^2 - 2 \\ \hline & 2x^2 & - 5x & + 6\\ & \phantom{-}-2x^2 & & + 4 & & & & \\ \hline & & -5x &+10 \\ \hline \end{array}$$
Here, Quotient = $$-x^2 - 2$$ and Remainder = $$-5x + 10$$.