The polynomial p(z) = z^2 - 2z + 5 has complex roots. Find both roots and verify they are conjugates.

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Multiple Choice

The polynomial p(z) = z^2 - 2z + 5 has complex roots. Find both roots and verify they are conjugates.

Real coefficients imply that if a non-real root appears, its complex conjugate is also a root. For p(z) = z^2 − 2z + 5, use the quadratic formula: z = [2 ± sqrt(4 − 20)]/2 = [2 ± sqrt(-16)]/2 = [2 ± 4i]/2 = 1 ± 2i. So the roots are 1 + 2i and 1 − 2i, which are conjugates because they have the same real part and opposite imaginary parts. This also checks with the sum of the roots being 2 and the product being 5, since (1+2i) + (1−2i) = 2 and (1+2i)(1−2i) = 1 + 4 = 5.

Real coefficients imply that if a non-real root appears, its complex conjugate is also a root. For p(z) = z^2 − 2z + 5, use the quadratic formula: z = [2 ± sqrt(4 − 20)]/2 = [2 ± sqrt(-16)]/2 = [2 ± 4i]/2 = 1 ± 2i. So the roots are 1 + 2i and 1 − 2i, which are conjugates because they have the same real part and opposite imaginary parts. This also checks with the sum of the roots being 2 and the product being 5, since (1+2i) + (1−2i) = 2 and (1+2i)(1−2i) = 1 + 4 = 5.

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