Question

Write a polynomial f(x) that satisfies the given conditions, Express the polynomial with the lowest possible leading positive integer coefficient

Polynomial of lowest degree with lowest possible integer coefficients, and with zeros 5-8i and 0 (multiplicity 4).

Answer 1

9514 1404 393

Answer:

f(x) = x^6 -10x^5 +89x^4

Explanation:

For a given zero p with multiplicity n, one of the factors of the polynomial will be (x -p)^n. If the polynomial has real coefficients, then the complex zeros come in conjugate pairs.

The factored form of your polynomial is ...

f(x) = (x -(5 -8i))(x -(5 +8i))(x -0)^4

f(x) = ((x -5)^2 -(8i)^2)(x^4) = (x^2 -10x +89)(x^4)

f(x) = x^6 -10x^5 +89x^4