Final answer:
The x-intercepts of the function f(x) can be found by factoring and setting f(x) to zero. One factor is already provided, (x - 5), and the remaining factors can be found through polynomial division. The y-intercept is found by evaluating f(0), which is the constant term in the polynomial.
Step-by-step explanation:
To find the x-intercepts and the y-intercept of the graph of f(x) = x³ - 7x² + 7x + 15 without using technology, we can use factoring and evaluation techniques. Given that (x - 5) is a factor of f(x), we can perform polynomial division or use synthetic division to find the other factors. To find the x-intercepts, we set f(x) = 0 which results in three factors (x - 5) being one of them. Setting the remaining quadratic equation equal to zero will yield the remaining x-intercepts. For the y-intercept, we evaluate f(0), which directly gives the y-intercept value, as it represents the point where the graph crosses the y-axis.
The function factored would look something like this f(x) = (x - 5)(ax² + bx + c), where a, b, and c are constants determined from the division process. Solving for x-intercepts involves finding the roots of the equation by setting the factored form equal to zero and solving for x. The y-intercept is the constant term from the original polynomial, which is 15 in this case.