domain of function in interval notation

Question

f(x) ={1x^{3}-13 x^{2}+26 x+112}/{1 x^{3}+10 x^{2}-63 x-648}

Answer 1

First of all let's rewrite it with fewer parentheses to make things easier.

f(x) = (x³ - 13x² + 26x + 112) / (x³ + 10x² - 63x - 648)

The best way to find the domain is to find values that are absent from it first.

Those numbers with satisfy x³ + 10x² - 63x - 648 = 0.

You can factor it using the rational root test. The candidates are

±1, ±2, ±3, ±4, ±6, ±8, ±9, ±12, ±18, ±24, ±27, ±36, ±54, ±72, ±81, ±108, ±162, ±216, ±324, ±648.

Checking each one shows that x = 8 and x = -9 both work.

You can factor to get (x - 8)(x + 9)² = 0 so there are no other solutions.

Thus the domain is (-∞,-9) U (-9,8) U (8,+∞).