Here’s how to factor the numerator $x^3 + x^2 + 2x + 1$ step-by-step:

Step 1: Group terms

Start by grouping the terms into two pairs:

$x^3 + x^2 + 2x + 1 = (x^3 + x^2) + (2x + 1)$

Step 2: Factor out common terms from each group

Now, factor out the greatest common factor (GCF) from each group of terms:

• From the first group $(x^3 + x^2)$, factor out $x^2$: $x^3 + x^2 = x^2(x + 1)$
• From the second group $(2x + 1)$, there’s no common factor, so it remains as is: $2x + 1 = 2x + 1$

Now, rewrite the expression with these factored terms:

$x^3 + x^2 + 2x + 1 = x^2(x + 1) + (2x + 1)$

Step 3: Look for a common binomial factor

Notice that both terms $x^2(x + 1)$ and $(2x + 1)$ involve expressions that suggest factoring. Specifically, you want to factor out the binomial $(x + 1)$, which appears implicitly.

So, rewrite the expression to combine the factored terms:

$x^3 + x^2 + 2x + 1 = (x + 1)(x^2 + 1)$

Step 4: Check the factorization

Multiply $(x + 1)(x^2 + 1)$ to check if it matches the original expression:

$(x + 1)(x^2 + 1) = x(x^2 + 1) + 1(x^2 + 1) = x^3 + x + x^2 + 1 = x^3 + x^2 + 2x + 1$

This matches the original numerator exactly, so the factorization is correct.

Final Factored Form:

$x^3 + x^2 + 2x + 1 = (x + 1)(x^2 + 1)$

This simplification allowed us to cancel the $(x + 1)$ term in the denominator and work with the simpler expression $x^2 + 1$ when finding the limit.

Would you like further clarification on this factorization process?

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Here are 5 related questions:

1. How do you recognize common factors in a polynomial?
2. Why does grouping terms help in factorization?
3. What other factorization techniques can be used for polynomials of degree 3 or higher?
4. How can synthetic division be used to factor polynomials?
5. What is the importance of verifying factorizations by multiplying back?

Tip: Always verify your factorization by expanding back to the original expression—it helps catch any mistakes.