Find all the zeros of polynomial $x^{4} - 2 x^{3} - 7 x^{2} + 8 x + 12$ , if two of its zeros are -1 and 2.

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Solution

product of roots $= 12$
$- 1 \times 2 \times α \times β = 12 α β = - 6$

sum of the roots $= 2 α + β - 1 + 2 = 2 α + β = 1 β = - \frac{6}{α} α - \frac{6}{α} = 1 α^{2} - 6 = α α^{2} - α - 6 = 0 α^{2} - 3 α + 2 α - 6 = 0 α (α - 3) + 2 (α - 3) = 0 (α + 2) (α - 3) = 0 α = 3, - 2$

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x^{4} - 2 x^{3} - 7 x^{2} + 8 x + 12

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