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write a polynomail function of least degree with rational coefficients so that P(x)=0 has roots 3i,3-i?

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  • write a polynomail function of least degree with rational coefficients so that P(x)=0 has roots 3i,3-i?


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Answer #1 | 13/02 2014 07:21
If an equation P(x) = 0 with real coefficients, has a complex solution (a+bi), then has also its conjugate (a-bi) If P(x) has roots 3i and 3 - i, then it must have also -3i and 3 + i remember that if a is a root then the polynomial contains the factor (x - a) the minimal degree polynomial is then P(x) = (x - 3i)(x + 3i)(x - (3 - i))(x - (3 + i)) P(x) = x^4 - 6 x^3 + 19 x^2 - 54 x + 90
Answer #2 | 13/02 2014 07:27
[ x - 3i ] [ x - (3 - i) ] = 0 x² - (3 - i) x - (3i)x + (3i)(3 - i) = 0 x² - [ (3 - i) +(3i) ] x + 9i + 3 = 0 x² - [ 3 + 2i ] x + 3 (1 + 3i) = 0

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