To simplify the rational expression \(\frac{3x - 3}{x^2 - 2x + 1}\), we first factor both the numerator and the denominator.
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Factor the numerator:
The numerator \(3x - 3\) can be factored as: \[ 3(x - 1) \] -
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: \[ \frac{3(x - 1)}{(x - 1)^2} \]
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Cancel the common factor:
We can cancel one \(x - 1\) from the numerator and the denominator (as long as \(x \neq 1\)): \[ \frac{3}{x - 1} \] -
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 \(x = 1\).
- 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 \(x = 1\).