$$ 1

Question

$$ 1< $$ Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression.

UpStudy AI · Accepted Answer

We start with the rational expression
$$
\frac{x^2-1}{x-1}.
$$
Our goal is to simplify it and then to determine which values of $$ x $$ must be excluded from the domain.

---

**Step 1. Factor the numerator**

Notice that the numerator is a difference of two squares. We can write
$$
x^2-1=(x+1)(x-1).
$$

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**Step 2. Rewrite the expression**

Substitute the factored form of the numerator into the expression:
$$
\frac{x^2-1}{x-1}=\frac{(x+1)(x-1)}{x-1}.
$$

---

**Step 3. Cancel the common factor**

Since we have a common factor of $$(x-1)$$ in the numerator and denominator, we can cancel it provided that
$$
x-1\neq 0.
$$
That is, we require
$$
x\neq 1.
$$
After canceling, the simplified expression becomes:
$$
x+1.
$$

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**Step 4. Determine the domain of the simplified expression**

Although the expression simplifies to $$ x+1 $$, we must remember that the original rational expression was undefined when the denominator was zero. Hence, the value
$$
x=1
$$
must be excluded from the domain.

---

**Final Answer**

The simplified expression is
$$
x+1
$$
with the domain restriction that
$$
x\neq 1.
$$
The simplified expression is $$ x + 1 $$, and the number that must be excluded from the domain is $$ x = 1 $$.

Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression.

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