could someone help me understand this?

Question

Reduce the rational expression to lowest terms. If it is already in lowest terms, enter the expression in the answer box.


(x³-1)/(x²-2x+1)

Answer 1

(x³-1)/(x²-2x+1)

 

First look at the numerator (x³-1)

This is the difference of two cubes.

(1)³ = 1

The formula for the difference of two cubes is

a³-b³ = (a-b)(a² + ab + b²)

For this polynomial a=x, b=1

 

x³-1 = (x-1)(x² + x + 1)

 

Now look at the denominator x²-2x+1

x²-2x+1 = (x-1)² = (x-1)(x-1)

 

Putting the rational expression back together

 

[(x-1)(x² + x + 1)]/[(x-1)(x-1)]

Cancel the common factor (x-1)

 

(x²+x+1)/(x-1)