Question

How do you write a polynomial in standard form given zeros 1,5,3-i?

Answer 1

`x^4-12x^3+51x^2-70x+50=0`.

Explanation:

Complex roots occur in conjugate pairs. So, as `3-i` is a root, `3+i` is also a root.

The roots are 1, 5, `3-i and 3+i`.

The biquadratic having these as roots is in the form
`x^4-S_1 x^3 + S_2 x^2-S_3x+S_4=0`, where
`S_1=sum`(roots) = 12,
`S_2=sum`(products of the roots, taken two at a time) = 51,
`S_3=sum`(product of the roots, taken three at a time) = 70
and `S_4=`product of all the roots =50.

Alternative method:

The factors of the polynomial are `x-1, x - 5, ((x-(3-i)) and (x-(3+i))`.
The biquadratic equation is

`(x-1)(x-3)((x-3)^2+1)=0`.

Expand and write the terms, in the order of descending powers of x.
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