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(i) deg p(x) = deg q(x)

(ii) deg q(x) = deg r(x)

(iii) deg r(x) = 0

Answer :

(i) p(x) = \(2x^2 + 2x + 8\)

=>q(x) = \(x^2 + x + 4\)

=> g(x) = \(2\) and r(x) = \(0\)

(ii) p(x) = \(x^3 + x^2 + x + 1\)

=>q(x) = \(x + 1\)

=> g(x) = \(x^2 - 1\) and r(x) = \(2x + 2\)

(iii) p(x) = \(x^3 - x^2 + 2x + 3\)

=>q(x) = \(x^2 + 2\)

=> g(x) = \(x - 1\) and r(x) = \(5\)

- Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following. (i) \(p(x) = x^3 - 3x^2 + 5x - 3 , g(x)= x^2 - 2\) (ii) \(p(x) = x^4 - 3x^2 + 4x + 5 , g(x)= x^2 - x + 1\) (iii) \(p(x) = x^4 - 5x + 6 , g(x)= 2 - x^2\)
- Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial. (i) \(t^2 - 3 , 2t^4 + 3t^3 -2t^2 - 9t -12\) (ii) \( x^2 + 3x + 1 , 3x^4 + 5x^3 - 7x^2 + 2x + 2\) (iii) \(x^2 - 3x + 1 , x^5 - 4x^3 + x^2 + 3x + 1\)
- Obtain all other zeroes of \(3x^4 + 6x^3 - 2x^2 -10x - 5\) if two of its zeroes are \(\sqrt{{5}\over{3}}\) and \(-\sqrt{{5}\over{3}}\).
- 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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