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Answer :
Final answer:
To find all zeros of the polynomial f(x), given three of its zeros, we deduce that the remaining zeros are the conjugates of the given complex zeros, since complex zeros appear in conjugate pairs for polynomials with real coefficients.
Explanation:
To find all the zeros of the function f(x)=x⁵-2x⁴-13x³+28x²+46x-60, given that 3-i, 13-i, and 1 are its zeros, we need to use the fact that complex zeros in polynomials with real coefficients come in conjugate pairs. This means that if 3-i is a zero, its conjugate 3+i is also a zero, and if 13-i is a zero, then its conjugate 13+i must be a zero as well. Thus, we have five zeros in total: 1, 3-i, 3+i, 13-i, and 13+i. Since the polynomial is of the fifth degree, this accounts for all the zeros. To validate these zeros or to further factorize the polynomial if needed, one can perform polynomial division or use synthetic division.
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