One factor of the function f(x) = x3 − 7x2 + 14x − 8 is (x − 4). describe how to find the x-intercepts and the y-intercept of the graph of f(x) without using technology. show your work and include all intercepts in your answer.

The x-intercepts are at (4,0), (2,0), and (1,0) and the y-intercept is at (0,-8).

Step-by-step explanation:

The cubic function is f(x) = x³ - 7x² + 14x - 8 (1)

Now, one factor of the function is (x - 4).

Then the other factors will be obtained as

x³ - 7x² + 14x - 8

= x³ - 4x² - 3x² + 12x + 2x - 8

= x²(x - 4) - 3x(x - 4) + 2(x - 4)

= (x - 4)(x² - 3x + 2)

= (x - 4)(x - 1)(x - 2)

Therefore, f(x) = (x - 4)(x - 1)(x - 2)

Now, for x-intercept f(x) = 0 = (x - 4)(x - 1)(x - 2)

⇒ x = 4 or x = 2 or x = 1

Therefore, the x-intercepts are at (4,0), (2,0), and (1,0).

Again, for y-intercept, x = 0, then from equation (1) we get,

f(0) = - 8

So, y-intercept is at (0,-8). (Answer)