Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each given of the following :

Question : Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each given of the following :

(i)  p(x) = \(x^3 – 3x^2 + 5x – 3\),  g(x) = \(x^2 – 2\)

(ii)  p(x) = \(x^4 – 3x^2 + 4x + 5\), g(x) = \(x^2 + 1 – x\)

(iii)  p(x) = \(x^4 – 5x + 6\),  g(x) = \(2 – x^2\)

Solution :

(i)  Here, dividend and divisor are both in standard forms. So, we have :

\(\therefore\)  The quotient is x – 3 and the remainder is 7x – 9

(ii)  Here, dividend is already in standard form and the divisor is not in the standard form. It can be written as \(x^2 – x + 1\). We have :

\(\therefore\)  The quotient is \(x^2 + x – 3\) and the remainder is 8.

(iii) Here, dividend and divisor are both in standard forms. So, we have :

\(\therefore\)  The quotient is \(-x^2 – 2\) and the remainder is -5x + 10.

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