Question

On dividing p(x) by a polynomial x-1-x2 the quotient and remainder were x-2 and 3 god p(x)

Answer 1

${\huge{\purple{\mathbb{\bold{ANSWER}}}}}$

▪︎ By Euclids Division Lemma we get,

=》 a = bq+r. --(1)

where a = divident, b = Divisor, q = Quotient, r = Remainder.

${\large{\blue{\bold{\underline{Given:-}}}}}$

▪︎ Divident = p(x).

▪︎ Divisor = x²-x-1.

▪︎ Quotient = x-2.

▪︎ Remainder = 3.

${\large{\blue{\bold{\underline{Now,}}}}}$

• By putting the above values in eq(1) we get,

$\implies \: p(x) = ( {x}^{2} - x - 1) \\ \\ (x - 2) + 3. \\ \\ \implies \: p(x) = {x}^{3} - {x}^{2} - x \\ \\ - 2 {x}^{2} + 2x + 2 + 3. \\ \\ (by \: \: opening \: \: brackets.) \\ \\ \i<strong>mplies \: </strong><strong>\</strong><strong>bold</strong><strong> </strong><strong>p(x)</strong><strong> </strong><strong>\</strong><strong>\</strong><strong> </strong><strong>\</strong><strong>\</strong><strong> </strong><strong>\</strong><strong>bold</strong><strong> </strong><strong>{</strong><strong>= {x}^{3} - 3 {x}^{2} + x + 3.</strong><strong>}</strong>$

☆ Therefore the p(x) is x³-3x²+x+3.

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