When f(x) = 4x³ − 8x² + 8x + 1 is divided by a polynomial g(x), we get (2x − 1) as quotient and (x + 3) as remainder. Find g(x).

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Solution

$\begin{array}{l} Let f (x) = (4 x^{3} - 8 x^{2} + 8 x + 1), q (x) = (2 x - 1) and r (x) = (x + 3) . \\ Then, f (x) = g (x) \times q (x) + r (x) \\ = > g (x) = \frac{f (x) - r (x)}{q (x)} \\ Now, [f (x) - r (x)] = (4 x^{3} - 8 x^{2} + 8 x + 1) - (x + 3) \\ = (4 x^{3} - 8 x^{2} + 7 x - 2) \\ On dividing (4 x^{3} - 8 x^{2} + 7 x - 2) by (2 x - 1), we get g (x) . \end{array}$

$\begin{array}{l} ∴ g (x) = 2 x^{2} - 3 x + 2 \end{array}$

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