To find the quotient of the polynomial x³ + 3x² + 3x + 2 divided by x² + x + 1, we can use polynomial long division.
First, we divide the leading term of the numerator x³ by the leading term of the denominator x². This gives us x.
Next, we multiply the entire divisor x² + x + 1 by x:
- x * (x² + x + 1) = x³ + x² + x
Now, we subtract this result from the original polynomial:
- (x³ + 3x² + 3x + 2) – (x³ + x² + x) = (3x² – x²) + (3x – x) + 2 = 2x² + 2x + 2
Next, we take the new polynomial 2x² + 2x + 2 and repeat the process. Divide the leading term 2x² by x², which gives us 2.
We then multiply the divisor by 2:
- 2 * (x² + x + 1) = 2x² + 2x + 2
Subtracting this from 2x² + 2x + 2 results in:
- (2x² + 2x + 2) – (2x² + 2x + 2) = 0
Since we are left with a remainder of 0, we can conclude that the quotient of the division is:
x + 2
Thus, the answer to the question is that the quotient of x³ + 3x² + 3x + 2 divided by x² + x + 1 is x + 2.