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Identify all zeros of the polynomial function
f(x) = x^{4} - 9x^{3} + 24x^{2} - 6x - 40.
a) -1, 4, 2 - i, 2 + i
b) -2, 4, 1 - i, 1 + i
c) -2, 4, 2 - i, 2 + i
d) -1, 4, 3 - i, 3 + i
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1. You can check manually that -1 is the root; 4 is the root; -2 is not the root.
This check is done by substitution of the values into the polynomial.
Thus the lines b) and c) are excluded, and only the lines a) and d) remain questionable.
2. Having two real roots (-1) and 4, you can divide the given polynomial by
(x-(-1)) = (x+1), and by
(x-4).
This division is division without remainder.
Then you get the quotient p(x) = .
It is the polynomial of the degree 2: quadratic polynomial with integer coefficients.
Then you can find the roots of this polynomial using the quadratic formula.
These roots will be complex numbers, perhaps.
Then you can compare these complex roots with that in lines a) and d).