Final answer:
To write the polynomial in factored form, we can use the rational root theorem to find the possible rational roots, and then divide the polynomial by one of the roots. The factored form of the polynomial is (x - 5)(x² - 2x - 35). To plot the function, find the x-intercepts by setting y = 0 and solve for x. Plot these points and draw the graph of the function.
Step-by-step explanation:
The given function is f(x) = x³ - 7x² - 25x + 175. To write it in factored form, we need to find the factors of the polynomial. The polynomial does not have any common factors, so we can use the rational root theorem to find the possible rational roots. By trying different values of x, we find that x = 5 is a root. Using long division, we can divide the polynomial by (x - 5) to get the factored form: f(x) = (x - 5)(x² - 2x - 35).
To plot the function and the points that fit on the axes, we can set y = 0 and solve for x to find the x-intercepts. Setting (x - 5)(x² - 2x - 35) = 0, we find x = 5 and x = 7 as the x-intercepts. Plot these points and draw the graph of the function to complete the plot.