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

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

p(x) = \(x^3 – 3x^2 + x + 2\)

q(x) = x – 2 and r(x) = -2x + 4

Solution : By division algorithm, we know that

p(x) = q(x) \(\times\) g(x) + r(x)

Therefore, \(x^3 – 3x^2 + x + 2\) = (x – 2) \(\times\) g(x) + (-2x + 4)

\(\implies\) \(x^3 – 3x^2 + x + 2 + 2x – 4\) = (x – 2) \(\times\) g(x)

\(\implies\)  g(x) = \(x^3 – 3x^2 + 3x – 2\over x – 2\)

On dividing  \(x^3 – 3x^2 + x + 2\) by x – 2, we get g(x)

polynomial image

Hence, g(x) = \(x^2 – x + 1\).

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