Check whether x + 1 is a factor of x³ + 3x² + 3x + 1 or not.

Let p(x) = x^{3 +} 3x^{2 +} 3x + 1

If after dividing x^{3 +} 3x^{2 +} 3x + 1 by x + 1 the remainder is 0 then x + 1 is a factor of x^{3 +} 3x^{2 +} 3x + 1

Now we will find remainder using remainder theorem which states that

If a polynomial p(x) is divided by a polynomial x – a then the remainder is p(a)

x + 1 can be written as x – (-1)

compare x – (-1) with x – a we get a = -1

⇒ p(a) = p(-1)

Substitute -1 in x^{3 +} 3x^{2 +} 3x + 1

⇒ p(-1) = (-1)^{3 +} 3(-1)^{2 +} 3(-1) + 1

⇒ p(-1) = -1 + 3 – 3 + 1

⇒ p(-1) = 0

As the remainder is 0 hence x + 1 is a factor of x^{3 +} 3x^{2 +} 3x + 1