find the quotient and remainder obtained on dividing p(x) = 4x^4 + 11x^3 + 2x^2 - 11x - 6 by x^2 + 2x + 2 and verify remainder by using remainder theorem
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On dividing p(x) = 4x^4 + 11x^3 + 2x^2
- 11x - 6 by g(x) = x^2 + 2x + 2
We get,
q(x) = 4x^2 + 3x – 12
r(x) = 7x + 18
Using remainder theorem,
g(x) x q(x) + r(x)
= (x^2 + 2x + 2)( 4x^2 + 3x – 12) + 7x + 18
= (4x^4 + 3x^3 – 12x^2 + 6x^3 + 6x – 24 + 8x^2 + 6x – 24) + (7x + 18) = 4x^4 + 11x^3 + 2x^2 - 11x – 6
= p(x)
We get,
q(x) = 4x^2 + 3x – 12
r(x) = 7x + 18
Using remainder theorem,
g(x) x q(x) + r(x)
= (x^2 + 2x + 2)( 4x^2 + 3x – 12) + 7x + 18
= (4x^4 + 3x^3 – 12x^2 + 6x^3 + 6x – 24 + 8x^2 + 6x – 24) + (7x + 18) = 4x^4 + 11x^3 + 2x^2 - 11x – 6
= p(x)
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