I'm glad to hear that my previous explanation helped you find some of the zeros of the function!
To address your question about synthetic division and long division, they are indeed both methods of dividing polynomials, but they have some differences in terms of their application and purpose.
Long division is a more general and widely applicable method of polynomial division. It involves the step-by-step division process that you may be familiar with from elementary school. In the case of finding zeros of a polynomial, long division can be used to divide the given polynomial by a linear factor to obtain a quotient polynomial. This quotient polynomial can then be further factored to find the remaining zeros.
On the other hand, synthetic division is a simplified form of polynomial division that can only be used when dividing by a linear factor of the form (x - c), where c is a constant. Synthetic division is often used when the divisor is a linear factor, as it is a quicker and more efficient method than long division in these cases.
In your case, you attempted to use synthetic division with the root 5-2i, but synthetic division only works with real numbers. This is because the coefficients in the synthetic division process are added or subtracted, and the complex numbers do not have a natural addition or subtraction operation. Therefore, synthetic division cannot be directly used with complex numbers as divisors.
To overcome this limitation and find the remaining zeros, you can use the fact that complex roots always come in conjugate pairs. This means that if 5-2i is a root of the polynomial, then 5+2i must also be a root. By multiplying the two conjugate roots together, you can obtain a quadratic factor.
Once you have this quadratic factor, you can then use either synthetic division or long division to divide the original polynomial by the quadratic factor to obtain a remainder that is a quadratic polynomial. Solving this quadratic polynomial will give you the remaining zeros of the function.
I hope this helps clarify the differences between synthetic division and long division, and why synthetic division did not work in your case. If you have any further questions, please let me know!