find a formula for a fourth degree polynomial.

Question

find a formula for a fourth degree polynomial. f(x) with leading coefficient of -2 and zeros of x=-5, x=0, and x=3i. write in complete factored form

Answer 1

Note it must have 4 roots.  You are given 3, but one of them is 3i, a complex number.  When a complex number is a root, its conjugate will also be a root, so -3i is the fourth root.  For each root r1, r2, r3, ...,

 

P(x) = (x-r1)(x-r2)...

 

To get a leading coefficient of -2, just take -2P(x) = f(x).  Multiplying by -2 will not change the roots.

 

Final answer:

 

f(x) = -2(x+5)(x)(x-3i)(x+3i) 

Answer 2

y = -2(x-0)(x+5)(x-3i)(x+3i) is the desired answer, with four linear factors, each of form (x - zero.of.the.function).

 

For the given zero "3i" we know that -3i is also a zero since complex roots occur in conjugate pairs (for a polynomial having all Real coefficients).