If a polynomial f(x) over the real number has the Complex roots (2+i) and (1-i), the f(x) could be?
- x⁴ - 6x³ + 15x² - 18x + 10
- x⁴ + 6x² + 10
- x⁴ - 7x² + 10
- None of these
Explanation
Real polynomial => complex roots come in conjugate pairs
So roots = 2+i, 2-i, 1-i, 1+i
Pair them:
(x - 2 - i)(x - 2 + i) = (x-2)^2 - i^2 = x^2 - 4x + 5
(x - 1 - i)(x - 1 + i) = (x-1)^2 - i^2 = x^2 - 2x + 2
Multiply:
(x^2 - 4x + 5)(x^2 - 2x + 2) = x^4 - 6x^3 + 15x^2 - 18x + 10
Last verified on 05-06-2026
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