Question 678195: Find a polynomial f(x) of degree 3 with real coefficients and the following zeros: -2, 3+i
Thanks for your help!
Answer by nerdybill(7384)   (Show Source):
You can put this solution on YOUR website! Find a polynomial f(x) of degree 3 with real coefficients and the following zeros: -2, 3+i
.
If a zero has an "imaginary" root such as:
3+i
then, there must be another imaginary root that is the conjugate:
3-i
.
so, all the zeros are:
-2, 3+i , 3-i
.
The factors must be:
(x - (-2)) , (x - (3+i)) , (x - (3-i))
(x+2) , (x-3-i) , (x-3+i)
.
Multiply the factors together to get the polynomial:
(x+2)(x-3-i)(x-3+i)
(x+2)(x(x-3+i)-3(x-3+i)-i(x-3+i))
(x+2)((x^2-3x+xi)-(3x-9+3i)-(xi-3i+i^2))
(x+2)((x^2-3x+xi)-(3x-9+3i)-(xi-3i-1))
(x+2)(x^2-3x+xi-3x+9-3i-xi+3i+1)
(x+2)(x^2-6x+xi+9-3i-xi+3i+1)
(x+2)(x^2-6x+9-3i+3i+1)
(x+2)(x^2-6x+9+1)
(x+2)(x^2-6x+10)
x(x^2-6x+10)+2(x^2-6x+10)
(x^3-6x^2+10x)+(2x^2-12x+20)
x^3-6x^2+10x+2x^2-12x+20
x^3-4x^2+10x-12x+20
x^3-4x^2-2x+20
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