To find the quotient when dividing the polynomial x³ + 3x² + 3x + 2 by x² + x + 1, we can use polynomial long division.
The first step is to divide the leading term of the numerator, x³, by the leading term of the denominator, x², which gives us x. We then multiply the entire divisor (x² + x + 1) by this quotient term:
x(x² + x + 1) = x³ + x² + x
Next, we subtract this result from the original polynomial:
(x³ + 3x² + 3x + 2) – (x³ + x² + x) = (3x² – x²) + (3x – x) + 2 = 2x² + 2x + 2
This new polynomial, 2x² + 2x + 2, is our new numerator. Now, we repeat the process: we divide 2x² by x², which gives us 2. We multiply the divisor by this number:
2(x² + x + 1) = 2x² + 2x + 2
We subtract this from our current numerator:
(2x² + 2x + 2) – (2x² + 2x + 2) = 0
Since we have a remainder of 0, the division is exact. Thus, the complete quotient when dividing x³ + 3x² + 3x + 2 by x² + x + 1 is:
x + 2
In conclusion, the answer to the question is that the quotient is x + 2.