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Answer :

Here, p(x) = \(x^3 - 3x^2 + x + 2\),

q(x) = \(x-2\),

r(x) = \(-2x+4\)

=>According to polynomial division algorithm,

=>p(x) = g(x).q(x) + r(x)

=>\(x^3 - 3x^2 + x + 2\) = g(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).\((x-2)\)

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

Solving it using long division:

\(\begin{array}{rrrr|ll} x^3 & -3x^2 & +3x & -2 & x - 2 \\ -x^3 & +2x^2 & & & x^2 - x + 1 \\
\hline & -x^2 & +3x & -2\\ \phantom{-} & +x^2 & -2x & & & \\ \hline & & x & -2 \\ & & -x & + 2 \\ \hline & & & 0\\ \hline \end{array}\)

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

- 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 \(-\sqrt{{5}\over{3}}\).
- Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm and (i) deg p(x) = deg q(x) (ii) deg q(x) = deg r(x) (iii) deg r(x) = 0

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