Understanding the Quotient of Polynomials
To find the quotient of the polynomial expression x³ + 3x² + 3x + 2 divided by x² + x + 1, we will use polynomial long division.
Step 1: Set Up the Division
We want to divide:
x³ + 3x² + 3x + 2
Step 2: Divide the Leading Terms
First, we look at the leading term of the dividend (x³) and the leading term of the divisor (x²). We divide the leading terms:
x³ ÷ x² = x
Step 3: Multiply and Subtract
Now, we multiply the entire divisor by x and subtract this from the original dividend:
x * (x² + x + 1) = x³ + x² + x
Now subtract:
(x³ + 3x² + 3x + 2) – (x³ + x² + x)
This leads to:
2x² + 2x + 2
Step 4: Repeat the Process
Now, we divide the new polynomial (2x² + 2x + 2) by x² + x + 1.
2x² ÷ x² = 2
Next, multiply by the divisor:
2 * (x² + x + 1) = 2x² + 2x + 2
Subtract again:
(2x² + 2x + 2) – (2x² + 2x + 2) = 0
Step 5: Conclusion
Since we end up with a remainder of 0, we conclude that:
x³ + 3x² + 3x + 2 ÷ x² + x + 1 = x + 2
Therefore, the quotient of the given polynomial expression is x + 2.