Question

The polynomial which when divided by $-x^{2}+x-1$ gives a quotient x-2 and remainder 3, is
(a) $x^{3}+3x^{2}+3x-5$
(b) $-x^{3}-3x^{2}-3x-5$
(c) $-x^{3}+3x^{2}-3x+5$
(d) $x^{3}-3x^{2}-3x+5$

Answer 1

SOLUTION :  

The correct option is (c) : -x³ +3x² - 3x +5 .

DIVISION ALGORITHM for polynomials :  

f(x) = g(x)  × q(x) + r(x)

Dividend =  Divisor ×  Quotient + Remainder

Given : g(x) = -x² + x -1 , q(x) = x - 2 and r(x) = 3

f(x) = g(x)  × q(x) + r(x)

= (-x² + x -1) (x - 2) + 3  

= -x³ + x² - x +2x² -2x + 2 + 3  

= -x³ + x² + 2x² - x -2x +2+3

= -x³ + 3x² - 3x +5

Hence, the polynomial f(x) is -x³ +3x² - 3x +5 .

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