Question

Write a polynomial in factored form with least possible degree having zeros at 5, -2 multiplicity 3, -3 multiplicity 2, 3i multiplicity 2, and f(-1)=-800.

Answer 1

Polynomial:
`f(x)=(x+1)(x-5)(x+6)^(2)(x-6i) - 800`

Explanation:

Zeros are at 5, (-2 * 3 = -6), (-3 * 2 = -6), (3i * 2 = 6i)

= 5, -6, -6, and 6i

The factors are (x-5), (x-(-6)), (x-(-6)), (x-6i)

= (x-5), (x+6), (x+6), (x-6i)

For f(-1) = -800, x = -1 which gives a factor of (x - (-1)) = x + 1. The remainder is -800

The polynomial is thus:

f(x) = (x+1)(x-5)(x+6)(x+6)(x-6i) - 800

`f(x)=(x+1)(x-5)(x+6)^(2)(x-6i) - 800`