Find the polynomial function that has integer coefficients, a leading coefficients of 1, and the least degree for the given zeros -1,2, and -i

Answer 1

To find a polynomial with integer coefficients, a leading coefficient of 1, and the least degree for the zeros −1−1, 22, and −i−i, we also need to include the conjugate ii.

The zeros are:

1. −1−1
2. 22
3. −i−i
4. ii

The polynomial is formed by multiplying the factors corresponding to these zeros:

f(x)=(x+1)(x−2)(x+i)(x−i)f(x)=(x+1)(x−2)(x+i)(x−i)

Simplify the complex conjugate pairs:

(x+i)(x−i)=x2+1(x+i)(x−i)=x2+1

Now, the polynomial is:

f(x)=(x+1)(x−2)(x2+1)f(x)=(x+1)(x−2)(x2+1)

Expand and combine like terms:

f(x)=(x2−x−2)(x2+1)=x4−x3−x2−x−2f(x)=(x2−x−2)(x2+1)=x4−x3−x2−x−2

So, the polynomial function is:

f(x)=x4−x3−x2−x−2f(x)=x4−x3−x2−x−2

This is the polynomial with integer coefficients, a leading coefficient of 1, and the least degree for the given zeros.