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Synthetic Division of Polynomials

A polynomial ...

Question

A polynomial p(x) is divided by (2x - 1). The quotient and remainder obtained are $(7 x^{2} + x + 5)$ and 4 respectively. Find p(x).
OR
Find the quotient and remainder using synthetic division.
$(3 x^{3} - 2 x^{2} + 7 x - 5) \div (x + 3)$ .

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Solution

Given that:
$q (x) = 7 x^{2} + x + 5, g (x) = 2 x - 1$ and $r (x) = 4$
Now,
$p (x) = g (x) . q (x) + r (x)$
$= (2 x - 1) . (7 x^{2} + x + 5) + 4$
$= 14 x^{3} + 2 x^{2} + 10 x - 7 x^{2} - x - 5 + 4$
$= 14 x^{3} - 5 x^{2} + 9 x - 1$
$∴ p (x) = 14 x^{3} - 5 x^{2} + 9 x - 1$

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