Algebra HW Help

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 6-5i and 0 (multiplicity 2).

Find f(x)

Answer 1

imaginary zeros come in conjugate pairs

if 6-5i is a zero, then 6+5i is also a zero

factors are (x-6+5i)(x-6-5i)

multiply them together

x=6-5i

x-6 = 5i

(x-6)^2 =-25

x^2 -12x+36+25 = 0

x^2 -12x +61= 0

x = 12/2 + or- (1/2)sqr(144-4(61)

x=6+ or- 5i are two zeros

(x^2-12x +61) is a factor

another factor is (x-0)^2 = x^2

multiply them together

x^2(x^2-12x +61)

= x^4 -12x^3 +61x^2 is the polynomial of least degree with the given zeros