Emma V.

asked • 03/02/17

Write a polynomial function of least degree that has rational coefficients, a leading coefficient of 1, and the zeros 1,3, and 2−i

I have no idea how to do this 

Kenneth S.

This question should not be given until a class lecture (or lectures) first precedes the assignment.
 
A key fact is that in this case the zeros can occur in complex conjugate pairs.
 
The polynomial can be written as a product of factors such as (x - root) involving the four roots (zeros).
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03/02/17

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Andrew M. answered • 03/02/17

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Mathematics - Algebra a Specialty / F.I.T. Grad - B.S. w/Honors

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If x=1 is a root then (x-1) is a factor
If x=3 is a root then (x-3) is a factor
If 2-i is a root, then so is 2+i
 
p(x) = (x-1)(x-3)(x-2+i)(x-2-i)
= (x2 - 4x +3)(x2-2x-ix-2x+4+2i+ix-2i-i2)
= (x2-4x+3)(x2-4x+5)
=(x4-4x3+5x2-4x3+16x2-20x+3x2-12x+15)
= x4-8x3+24x2-32x+15
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