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.

x-intercepts → x = 1, 2 and 4

y - intercept → y = -8

Step-by-step explanation:

Given function is f(x) = x³ - 7x² + 14x - 8

Known factor of this function is (x - 4)

For x - intercept we substitute f(x) = 0 in the function and for y-intercept we substitute x = 0

For x - intercepts,

x³ - 7x² + 14x - 8 = 0

Since (x - 4) is a factor of this equation,

By synthetic division,

4) 1    -7    14     -8

    -     4   -12      8      

   1     -3    2      0

Therefore, factored form of the equation will be,

(x - 4)(x² - 3x + 2) = 0

(x - 4)(x² - 2x - x + 2) = 0

(x - 4)[x(x - 2) - 1(x - 2)] = 0

(x - 4)(x - 1)(x - 2) = 0

Therefore, x = 1, 2 and 4 will be the x intercepts.

Similarly, for y-intercepts,

f(x) = 0 - 7(0) + 14(0) - 8

     = -8

Therefore, y-intercept is y = -8