Final answer:
To find all zeros of the polynomial f(x)=x^3-15x^2+65x-51 given that x-1 is a factor, we first divide the polynomial by x-1, then solve the resulting quadratic equation using the quadratic formula.
Step-by-step explanation:
We are asked to find all zeros of the polynomial function f(x)=x^3-15x^2+65x-51, given that x-1 is a factor of f(x). The first step in finding the zeros is to divide the polynomial by x-1 to find the other factors.
Using synthetic division or long division, we can find the quotient polynomial. Next, we find the zeros of the quotient polynomial, which is a quadratic. It will be of the form ax^2+bx+c. We can solve for zeros using the quadratic formula, x = (-b ± √(b^2-4ac))/(2a).
After determining all zeros from this quadratic, we will have the complete set of real and complex zeros of the original cubic polynomial, along with the zero from the factor x-1.