Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression in order for it to be equivalent to the original expression.

3x-3
x2-2x+1

=

Simplify the rational expression.

3x-3
x2-2x+1

(Simplify your answer. Use positive exponents only.)

1 answer

To simplify the rational expression \(\frac{3x - 3}{x^2 - 2x + 1}\), we first factor both the numerator and the denominator.

  1. Factor the numerator:
    The numerator \(3x - 3\) can be factored as: \[ 3(x - 1) \]

  2. Factor the denominator:
    The denominator \(x^2 - 2x + 1\) is a perfect square trinomial, and can be factored as: \[ (x - 1)^2 \]

Now the expression becomes: \[ \frac{3(x - 1)}{(x - 1)^2} \]

  1. Cancel the common factor:
    We can cancel one \(x - 1\) from the numerator and the denominator (as long as \(x \neq 1\)): \[ \frac{3}{x - 1} \]

  2. Determine the domain exclusions:
    The original expression \(\frac{3x - 3}{x^2 - 2x + 1}\) has the denominator \(x^2 - 2x + 1\). This denominator must not equal zero. We set the denominator equal to zero to find the points of exclusion: \[ x^2 - 2x + 1 = 0 \] Factoring gives: \[ (x - 1)^2 = 0 \] This results in \(x = 1\) being a double root (the only solution).

Thus, the number that must be excluded from the domain is \(x = 1\).

  1. Final simplified expression:
    The final simplification of the rational expression is: \[ \frac{3}{x - 1}, \quad x \neq 1 \]

In summary:

  • The simplified expression is \(\frac{3}{x - 1}\).
  • The excluded value from the domain is \(x = 1\).