Domain of a function and simplification of fractions

I was working on the following exercise from a math book:

Question: what is the domain of the following real function:

$$ f(x)= \frac{x-1}{(x^2-1)(x^2-7x+10)} $$

Step 1, I develop:

$$ \frac{x-1}{(x-1)(x+1)(x-2)(x-5)} $$

Step 2, I simplify:

$$ \frac{1}{(x+1)(x-2)(x-5)} $$

And I find the following answer:

$$ D_f = \mathbb{R}_{\setminus\{-1,2,5\}} $$

I then checked the correction in the book which gives the following correct answer:

$$ D_f = \mathbb{R}_{\setminus\{-1,1,2,5\}} $$

But if $1$ must be excluded from the domain of $f$, it would lead to a contradiction. In effect, $1$ comes from the $(x - 1)$ that I simplified in step 2. If I exclude $1$ from the domain on that basis, I may also multiply the numerator and denominator by any real $a$ of the form $\frac{x+a}{x+a}$ and state that $a$ must be excluded from the domain of $f$. In consequence, we could no longer use fractions in algebra!

I feel my answer is correct and the book is incorrect, but since this is improbable in view of my level in math, I would like someone to confirm this.

Thanks in advance for your help.