Question

How do you write a polynomial in standard form given the zeros x=2, -6, and 2+ 4i?

Answer 1

`x^4-8x^2+128x-240`

Explanation:

`"given the zeros "x=a,x=b,x=c,x=d`

`"then the factors are "(x-a),(x-b),(x-c),(x-d)`

`"the polynomial is then the product of the factors"`

`p(x)=(x-a)(x-b)(x-c)(x-d)`

`"here one of the given zeros is complex " 2+4i`

`"complex zeros always occur in "color(blue)"conjugate pairs"`

`rArr2-4i" is also a zero of the polynomial"`

`"the four zeros are "x=2,x=-6,x=2+-4i`

`"4 zeros indicate a polynomial of degree 4"`

`rArrp(x)=(x-2)(x+6)(x-2-4i)(x-2+4i)`

`color(white)(rArrp(x))=(x^2+4x-12)(x^2-4x+20)`

`color(white)(rArrp(x))=x^4-8x^2+128x-240" possible polynomial"`