1 What are the least and most amount of distinct zeroes of a

1. What are the least, and most, amount of distinct zeroes of a 7th degree polynomial, given that at least one root is a complex number?

2. What is f(x) = x⁸ - 1 divided by x - 1?

3. What are the zeroes of f(x) = x³ + 5x² - 7x + 1?

Solution

Since multiple problems are posted, I am answering the first one for you:)

Note: The complex zeros always occur in pairs. It means if a+ib is a zero, then automatically a-ib is also a zero. (It means there cannot be exactly 1 complex zero. which obviously means there cannot be odd number of complex zeros).

A 7th degree polynomial has atmost 7 zeros.

It is given that a zero of the 7th degree polynomial is a complex number. Since complex zeros occur in pairs(see note), this polynomial should have one more complex zero. So together there would be 2 complex zeros for sure.

So there can be maximum of 7 - 2 = 5 real zeros.

And the polynomial has atmost 6 complex zeros (in 3 pairs) and 1 real zero.

So together the polynomial can have

2 complex zeros and 5 real zeros (max) and

6 complex zeros and 1 real zero (min)