I don't even know where to start, teacher gave us this with no direction.

Suppose the polynomial R(x)=a9x^9+a8x^8+...+a,x+a0 has real coefficients with a9 cannot equal 0. Suppose also that R(x)has the following zeros:

5,
6,
5+5i.
Using this information, answer the following questions.

(a) What is another zero of R(x)?
(b) At most how many real zeros of R(x) are there?
(c) At most how many imaginary zeros of R(x) are there?

a) 5-5i they come in pairs

b) 9 total, two are complex conjugates so 7

c) we have accounted for four, two real, two are complex conjugates. That leaves 5.
One of those must be real because complex roots come in paira
that leaves 4
or at most two pairs.

To answer these questions, we need to understand the concept of polynomial zeros and their relationship with the coefficients of the polynomial. The Polynomial Remainder Theorem states that if a polynomial P(x) is divided by (x-a), where 'a' is a real or complex number, then the remainder will be zero if and only if 'a' is a root or zero of the polynomial.

Now, let's use the given information to find the answers to the questions:

(a) To find another zero of R(x), we can use the Conjugate Root Theorem. According to this theorem, if a polynomial with real coefficients has a complex root (in this case, 5+5i), then its conjugate (5-5i) will also be a root. Therefore, another zero of R(x) is 5-5i.

(b) To determine the maximum number of real zeros, we can apply the Fundamental Theorem of Algebra. It states that a polynomial of degree 'n' can have at most 'n' complex zeros. In this case, R(x) is a polynomial of degree 9, and it already has three zeros: 5, 6, and 5+5i. Hence, the maximum number of real zeros is 9 - 3 = 6.

(c) Using the same logic as part (b), we know that there can be at most 9 total zeros (real or complex) for R(x) since it is a polynomial of degree 9. We already have three known zeros (5, 6, and 5+5i), which leaves us with a maximum of 9 - 3 = 6 imaginary (complex) zeros.

In summary:
(a) Another zero of R(x) is 5-5i.
(b) At most, there can be 6 real zeros of R(x).
(c) At most, there can be 6 imaginary zeros (complex zeros) of R(x).