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Range of Quadratic Expression

Find all zero...

Question

Find all zeros of the polynomial $x^{4} + x^{3} - 34 x^{2} - 4 x + 120,$ if two of its zeros are $2$ and $- 2$ .

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Solution

roots $= a, b$
product of roots $= 120$
$a b \times - 4 = 120 a b = - 30 a = - \frac{30}{b} a + b + 2 - 2 = - 1 b - \frac{30}{b} = - 1 b^{2} + b - 30 = 0 b^{2} + 6 b - 5 b - 30 = 0 b (b + 6) (b - 5) = 0$
roots are $= 5, - 6$

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