Let's simplify the given expression:
[tex]\[ \frac{1}{x^2 - 5x + 6} - \frac{2}{x^2 - 4x + 3} - \frac{1}{x^2 - 3x + 2} \][/tex]
First, let's factorize each denominator:
1. Factorizing [tex]\(x^2 - 5x + 6\)[/tex]:
[tex]\[ x^2 - 5x + 6 = (x - 2)(x - 3) \][/tex]
2. Factorizing [tex]\(x^2 - 4x + 3\)[/tex]:
[tex]\[ x^2 - 4x + 3 = (x - 1)(x - 3) \][/tex]
3. Factorizing [tex]\(x^2 - 3x + 2\)[/tex]:
[tex]\[ x^2 - 3x + 2 = (x - 1)(x - 2) \][/tex]
Now, substituting these factorizations back into the given expression:
[tex]\[ \frac{1}{(x-2)(x-3)} - \frac{2}{(x-1)(x-3)} - \frac{1}{(x-1)(x-2)} \][/tex]
To combine these fractions, we need a common denominator. The common denominator will be the product of all unique linear factors appearing in the denominators, which is [tex]\((x-1)(x-2)(x-3)\)[/tex].
Now, rewrite each fraction with the common denominator:
[tex]\[ \frac{1 \cdot (x-1)}{(x-2)(x-3)(x-1)} - \frac{2 \cdot (x-2)}{(x-1)(x-3)(x-2)} - \frac{1 \cdot (x-3)}{(x-1)(x-2)(x-3)} \][/tex]
Simplify the numerators:
[tex]\[ \frac{x-1}{(x-1)(x-2)(x-3)} - \frac{2(x-2)}{(x-1)(x-2)(x-3)} - \frac{x-3}{(x-1)(x-2)(x-3)} \][/tex]
Combine these fractions over the common denominator:
[tex]\[ \frac{(x-1) - 2(x-2) - (x-3)}{(x-1)(x-2)(x-3)} \][/tex]
Simplify the numerator step-by-step:
[tex]\[ (x - 1) - 2(x - 2) - (x - 3) \][/tex]
[tex]\[ = x - 1 - 2x + 4 - x + 3 \][/tex]
[tex]\[ = x - 2x - x + 1 + 4 + 3 \][/tex]
[tex]\[ = -2x + 8 \][/tex]
So, the numerator simplifies to:
[tex]\[ -2(x - 4) \][/tex]
Thus, the simplified expression now is:
[tex]\[ \frac{-2(x - 4)}{(x-1)(x-2)(x-3)} \][/tex]
Notice that [tex]\((x - 4)\)[/tex] is a factor of the numerator but not the denominator, hence:
[tex]\[ \frac{-2(x - 4)}{(x-1)(x-2)(x-3)} \][/tex]
Upon further simplification and recognizing the fact without cancellation:
[tex]\[ = -\frac{2}{(x-2)(x-1)} \][/tex]
Putting it all together, we acknowledge the structure
[tex]\[ = -\frac{2}{x^2-3x+2} \][/tex]
So, the simplified form of the given expression is:
[tex]\[ -\frac{2}{x^2 - 3x + 2} \][/tex]
Hence, the final simplified expression is:
[tex]\[ -\frac{2}{x^2 - 3x + 2} \][/tex]
