Question 1071605: Find a polynomial function of least degree having only real coefficients, a leading coefficient of 1, and zeros of 4 and 1+I.
Answer by Edwin McCravy(20056)   (Show Source):
You can put this solution on YOUR website! Find a polynomial function of least degree having
only real coefficients, a leading coefficient of 1,
and zeros of 4 and 1+I.
x = 4 x = 1 + i x = 1 - i
x - 4 = 0 x - 1 - i = 0 x - 1 + i = 0
Multiply the three left sides together:
(x - 4)(x - 1 - i)(x - 1 + i)
and set that product equal to the product of the
three right sides (0)(0)(0)=0
(x - 4)(x - 1 - i)(x - 1 + i) = 0
(x - 4)[(x - 1) - i][(x - 1) + i] = 0
(x - 4)[(x - 1)² - i²] = 0
(x - 4)[(x - 1)² - (-1)] = 0
(x - 4)[(x - 1)² + 1] = 0
(x - 4)[x² - 2x + 1 + 1] = 0
(x - 4)(x² - 2x + 2) = 0
x³ - 2x² + 2x - 4x² + 8x - 8 = 0
x³ - 6x² + 10x - 8 = 0
Edwin
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