factor the polynomial completely

Question

x³ - 125

 

and 

 

x³ - 6x² - 9x + 54 

Answer 1

x³ - 125 = x³ - 5³

 

This is a difference of cubes which has a standard factoring formula:

 

a³ - b³ = (a-b)(a²+ab+b²)

 

In your case, a = x and b = 5, so we get:

 

x³ - 5³ = (x-5)(x² + 5x + 25)

 

which is as far as it can be factored.

 

-----

 

x³ - 6x² - 9x + 54

 

Factor by grouping:

 

(x³ - 6x²) - (9x - 54)

x²(x - 6) - 9(x - 6)

(x²-9)(x-6)

 

Now (x² - 9) is a "difference of squares" which factors to (x+3)(x-3).  So the final factored form is: 

 

(x+3)(x-3)(x-6)