Question

How do you write a polynomial function of least degree and leading coefficient 1 when the zeros are 5, 2+3i?

Answer 1

`p(x)=x^3-9x^2+33x-65`

Explanation:

`"by the factor theorem"`

`"if "x=a" is a zero then "`

`(x-a)" is a factor of the polynomial"`

`"note that complex zeros occur in conjugate pairs"`

`x=2+3i" is a zero then "2color(red)(-)3i" is also a zero"`

`p(x)=(x-5)(x-(2+3i))(x-(2-3i))`

`color(white)(p(x))=(x-5)(x-2-3i)(x-2+3i)`

`color(white)(p(x))=(x-5)((x-2)^2-(3i)^2)`

`color(white)(p(x))=(x-5)(x^2-4x+4+9)`

`color(white)(p(x))=(x-5)(x^2-4x+13)`

`color(white)(p(x))=x^3-9x^2+33x-65`