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

##### 1 Answer

# P(x) = (x+2)(x+3)(x^2+1) #

# " " = x^4+5x^3+7x^2+5x+6 #

#### Explanation:

Suppose the polynomial is

By the factor theorem, if

We have the following roots of

# x=-2,-3,i,-i #

Hence, the following are factors of

# (x+2), (x+3), (x-i), (x+i) #

Hencde, we can write the polynoimal of least degree as the product of these factors (any higher degree polynomial would have additional roots)

# P(x) = A(x+2)(x+3)(x-i)(x+i) #

We want our polynomial to have leading coefficient

# (x-i)(x+i) = x^2 + ix - ix -i^2 = x^2+1#

Thus we have:

# P(x) = (x+2)(x+3)(x^2+1) #

# " " = x^4+5x^3+7x^2+5x+6 #