Divide the polynomial f(x) = 3x² – x³ – 3x + 5 by the polynomial g(x) = x – 1 – x² and verify the division algorithm.

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Solution

Given,

f(x) = 3x² – x³ – 3x + 5

g(x) = x – 1 – x²

Dividing f(x) = 3x² – x³ – 3x + 5 by g(x) = x – 1 – x^2

So,

Here,

Quotient = q(x) = x – 2

Remainder = r(x) = 3

By division algorithm of polynomials,

Dividend = (Quotient × Divisor) + Remainder

So,

[q(x) × g(x)] + r(x) = (x – 2)(x – 1 – x²) + 3

= x² – x – x³ – 2x + 2 + 2x² + 3

= 3x² – x³ – 3x + 5

= f(x)

Hence, the division algorithm is verified.

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Divide the polynomial $f (x) = 3 x^{2} - x^{3} - 3 x + 5$ by the polynomial $g (x) = x - 1 - x^{2}$ and, verify the division algorithm.

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