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

1 Answer
Aug 13, 2017

Answer:

# P(x) = (x+2)(x+3)(x^2+1) #
# " " = x^4+5x^3+7x^2+5x+6 #

Explanation:

Suppose the polynomial is #P(x)#

By the factor theorem, if #x=a# is root of #P(x)=0#, then #x=a# is a factor of #P(x)#

We have the following roots of #P(x)=0#

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

Hence, the following are factors of #P(x)#

# (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 #1=>A=1#, and also to have real coefficients, so let us multiply out the complex factors (as they are conjugates we will get a real product)

# (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 #