Final answer:
To find all zeros of the polynomial f(x), we use the known zero of -7 to factor the polynomial and then apply the quadratic formula to the resulting quadratic equation to find the remaining zeros.
Step-by-step explanation:
If f(x) = x³ - 10x² + 17x - 28 and f(-7) = 0, the student is instructed to find all the zeros of f(x) algebraically. Since we know that f(-7) = 0, -7 is a zero of the polynomial. To find the other zeros, we can perform polynomial division or use synthetic division to divide the polynomial by (x + 7) to find the quadratic factor.
After long or synthetic division of f(x) by (x + 7), we should get a quadratic equation of the form ax² + bx + c = 0. Then, we can apply the quadratic formula to find the remaining zeros.
The quadratic formula is given by x = (-b ± √(b² - 4ac))/(2a). Substituting the corresponding a, b, and c values from the quadratic factor into this formula will yield the two other zeros of the polynomial.