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On dividing x^{3} – 3x^{2} + x + 2 by a polynomial g(x), the quotient and remainder were x – 2 and –2x + 4, respectively. Find g(x)

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#### Solution

p(x) = x^{3}-3x^{2}+x+2 (Dividend)

g(x) = ? (Divisor)

Quotient = (x - 2)

Remainder = (-2x + 4)

Dividend = Divisor × Quotient + Remainder

x^{3} - 3x^{2} + x + 2 = g(x) x (x - 2) + (-2x + 4)

x^{3} - 3x^{2} + x + 2 +2x - 4 = g(x)(x - 2)

x^{3} - 3x^{2} + 3x - 2 = g(x)(x - 2)

g(x) is the quotient when we divide (x^{3} -3x^{2} + 3x - 2) by (x - 2)

∴ g(x) = (x^2 - x + 1)

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