Question 652184: Find a polynomial function with real coefficients that has the given zeros.
3, 1-3i
Answer by nerdybill(7384)   (Show Source):
You can put this solution on YOUR website! Find a polynomial function with real coefficients that has the given zeros.
3, 1-3i
if you have a complex root, then you know you also have a root that is its conjugate: 1+3i
.
the roots are:
3, 1-3i, 1+3i
.
If these are the roots, then the factors must be:
(x-3), (x-(1-3i)), (x-(1+3i))
or
(x-3), (x-1+3i), (x-1-3i)
.
multiply all three factors to get your polynomial:
(x-3)(x-1+3i)(x-1-3i)
(x-3)(x(x-1-3i)-1(x-1-3i)+3i(x-1-3i))
(x-3)(x^2-x-3xi-x+1+3i+3xi-3i+9)
(x-3)(x^2-x-x+1+3i-3i+9)
(x-3)(x^2-x-x+1+9)
(x-3)(x^2-2x+10)
x^2(x-3)-2x(x-3)+10(x-3)
x^3-3x^2-2x^2+6x+10x-30
x^3-5x^2+6x+10x-30
x^3-5x^2+16x-30
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