Question

write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros.

1. -2,-1,1

2. i,4

3. i,2 -sqrt3

4. 1,4,1+sqrt2

Answer 1

A polynomial of degree n that has say 7 as a root, must be of the form
(x-7) g(x)
where g(x) is polynomial of degree n-1

Answer 2

A polynomial of degree n that has say 7 and -5 as a root, must be of the form
(x-7)(x+5) g(x)
where g(x) is polynomial of degree n-2

Answer 3

A polynomial that has 2 + 5 i as a root, has also 2-5i as a root.

Answer 4

A polynomial that has 2 + sqrt(3) as a root, has also 2- sqrt(3) as a root.

Answer 5

For 1) since -2, -1 and 1 as roots, then
your polynomial is
f(x)= a(x+2)(x+1)(x-1)
a must be 1 since the leading coef. is 1 so
f(x)= (x+2)(x+1)(x-1)

Answer 6

Do the remaining ones