Question

The graph of f(x) = 2x3 – 19x2 + 57x – 54 is shown below. How many roots of f(x) are rational numbers?

Answer 1

Answer:

3 roots. 2, 3 and 9/2. All are rational numbers.

Step-by-step explanation:

Find the root of the function by Zero function value.

f(1) = -14

f(2) = 0.

Thus 2 is a root of the above function.

Dividing f(x) by (x-2) we get

f(x) = 2x³ - 19x² + 57x - 54 = (x - 2)(2x² - 15x + 27)

Finding roots of the 2x² - 15x + 27 using formula we get

15 + √(225 - 216)/4   and 15 - √(225 - 216)/4

18/4 and 3

f(x) = (x - 2)(x - 3)(x - 9/2)

Thus roots of the above functions are 2, 3 and 9/2.