Find the remainder when $30 y^{4} + 11 y^{3} - 42 y^{2} + 3$ is divided by $3 y^{2} + 2 y$ ?

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Solution

Given: Here, we have to divide $30 y^{4} + 11 y^{3} - 42 y^{2} + 3$ by $3 y^{2} + 2 y i . e .$ $\frac{30 y^{4} + 11 y^{3} - 42 y^{2} + 3}{3 y^{2} + 2 y}$
$10 y^{2} - 3 y - 12$
$(3 y^{2} + 2 y) \sqrt{30 y^{4} + 11 y^{3} - 42 y^{2} + 3} 30 y^{4} + 20 y^{3} - - - ------------------ - - 9 y^{3} - 42 y^{2} + 3 - 9 y^{3} - 6 y^{2} + + - ------------------ - - 36 y^{2} + 3 - 36 y^{2} - 24 y + + - ------------------ - 24 y + 3 - ------------------ -$
$∴$ Remainder obtained when $30 y^{4} + 11 y^{3} - 42 y^{2} + 3$ is divided $3 y^{2} + 2 y$ is $24 y + 3$ .

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