Factor x^3 + x^2 + x + 1 by grouping: What is the resulting expression?

To factor the expression x³ + x² + x + 1 by grouping, we can start by rearranging it in pairs for easier factorization.

First, we can group the first two terms and the last two terms:

(x³ + x²) + (x + 1)

Now, we can factor out the common factors from each group:

• From the first group (x³ + x²), we can factor out x², leading to:

x²(x + 1)

• From the second group (x + 1), we can factor out 1 (as it does not change anything), leading to:

1(x + 1)

Putting it all together, we have:

x²(x + 1) + 1(x + 1)

Notice that (x + 1) is a common factor now, so we can factor that out:

(x + 1)(x² + 1)

Thus, the final factored form of the expression x³ + x² + x + 1 by grouping is:

(x + 1)(x² + 1)