What is the quotient when you divide the polynomial x³ + 3x² + 3x + 2 by x² + x + 1?

Understanding Polynomial Division

In polynomial division, we divide one polynomial by another, similar to how we would do with numbers. Here, we want to find the quotient of the polynomial x³ + 3x² + 3x + 2 by the polynomial x² + x + 1.

Step 1: Set Up the Division

We begin by setting up the long division:

	   _____________
	x² + x + 1 | x³ + 3x² + 3x + 2

Step 2: Divide the Leading Terms

We look at the leading terms of both polynomials. The leading term of x³ + 3x² + 3x + 2 is x³, and the leading term of x² + x + 1 is x².

Now, we divide x³ by x², which gives us x. This is our first term of the quotient.

Step 3: Multiply and Subtract

Next, we multiply x by x² + x + 1:

		x * (x² + x + 1) = x³ + x² + x

Now, we subtract this result from the original polynomial:

	   x³ + 3x² + 3x + 2
	- (x³ + x² + x)
	____________________
	       2x² + 2x + 2

Step 4: Repeat the Process

Now, we take the new polynomial 2x² + 2x + 2 and repeat the division process:

Divide the leading term 2x² by x², which gives us 2.

Now, we multiply 2 by x² + x + 1:

		x * (2) = 2x² + 2x + 2

Now, we perform the subtraction:

	   2x² + 2x + 2
	- (2x² + 2x + 2)
	____________________
	          0

Conclusion

Since there is no remainder, the quotient when dividing the polynomial x³ + 3x² + 3x + 2 by x² + x + 1 is:

		x + 2

So, the final answer is: