Question

A 4th-degree polynomial function has zeros at 3 and 5-i. Can 4+i also be a zero of the function? Please explain how.

Answer 1

hint
if a+bi s a zero of a real polynomial, what else has to be a zero?

Answer 2

I'm not too sure, but maybe
a−bi

Answer 3

yes
so if 5−i is a zero, what else has to be a zero?

Answer 4

5+i
?

Answer 5

right, so now you have three zeros
3,5+i,5−i
how many zeros can a 4th degree polynomial have?

Answer 6

A 4th-degree polynomial can have 4 zeros.

Answer 7

yea
so can 4+i be a zero?

Answer 8

well actually it could, but not if the polynomial has REAL coefficients, which is probably what is meant here

Answer 9

ok yeah, but if the coefficients are real, then if 4+i is a zero, so is 4−i and now they are too many

Answer 10

lol "there are too many"

Answer 11

Wait, do polynomial functions with complex coefficients have at least one complex zero?

Answer 12

fundamental theorem o' algebra says any polynomial has at least one zero

Answer 13

doesn't say if the zeros are real or complex though...

Answer 14

which means, counting multiplicity, a polynomial of degree n has n zeros

Answer 15

Okay, so I looked up the theorem, and a statement said: "Every polynomial function of degree
n≥1
has at least one complex zero."

Answer 16

true that, as the kids say

Answer 17

But I still don't understand. The polynomial function has real coefficients, so 4+i cannot be a zero because...?

Answer 18

this still doesn't really answer the question
i am going to guess that whoever wrote it, had in mind polynomials with real coefficients
you are not at the point to understand what a function with complex coefficients would mean, although i could be wrong

Answer 19

beause so would 4−i so you would have 5 zeros
3,5+i,5−i,4+i,4−i
one too many

Answer 20

So, because 5-i is a zero, 5+i is a zero. Then, 3 is a zero, along with another unknown zero. That zero cannot be 4+i because, if 4+i were a zero, then there would be a 4-i, which is one too many.