simplifying rational expresion

Question

3x²-3x

2x²-2x

Answer 1

Simplify f(x) = (3x^2 - 3x) / ( 2x^2 -2).

 

Later you will want to graph f(x), so we have to be careful about any "cancellations".

 

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

 

Factorize numerator and denominator

using Greatest Common Factor:

 

f(x) = 3x(x - 1) /(2(x^2 - 1))

 

Apply Difference of Squares:

 

f(x) = 3x(x - 1) /(2(x + 1)(x - 1))

 

(x - 1)/(x - 1) = {1 if x ≠ 1, UNDEFINED if x = 1}

When you graph f(x) you must indicate a "Hole" at x = 1.

 

f(x) = 3x/(2(x + 1)), x ≠ 1; The constraint MUST be indicated.

 

The hole is at (1,3/4).

 

f(x) = (3/2)x/(x + 1), x ≠ 1

 

To simplify further, do long division:

 

x     | 1

x+1

–––

0-1 = R

 

f(x) = (3/2)(1 – 1/(x + 1)), x ≠ 1

 

f(x) = (3/2) – 3/(2x + 2)), x ≠ 1

 

From this equation you can pick off:

 

Horizontal Asymptote: y = 3/2

 

Vertical Asymptote: x = –1

 

Hole: x = 1

 

Answer 2

(3X^2 - 3x) / ( 2X^2 -2) =

 

  3x( X -1 ) / 2( X +1) ( X-1)

 

     3x / ( 2X +2)