Understanding the Quotient of Polynomials
To find the quotient of the polynomial x³ + 3x² + 5x + 3 divided by x + 1, we can use polynomial long division. This process is similar to regular long division but applies to polynomials instead. Let’s break it down step-by-step:
Step 1: Set Up the Division
We want to divide x³ + 3x² + 5x + 3 by x + 1.
Step 2: Divide the Leading Terms
First, we take the leading term of the dividend (x³) and divide it by the leading term of the divisor (x), giving us x².
Step 3: Multiply and Subtract
Next, we multiply x² by the entire divisor (x + 1):
- x² * (x + 1) = x³ + x²
Then, we subtract this result from the original polynomial:
- (x³ + 3x² + 5x + 3) – (x³ + x²) = 2x² + 5x + 3
Step 4: Repeat the Process
Now, we divide the new leading term (2x²) by the leading term of the divisor (x), which gives us 2x.
We multiply 2x by the divisor:
- 2x * (x + 1) = 2x² + 2x
Subtract again:
- (2x² + 5x + 3) – (2x² + 2x) = 3x + 3
Step 5: Divide Again
Next, we take the leading term (3x) and divide it by (x), yielding 3.
Multiply and subtract:
- 3 * (x + 1) = 3x + 3
- (3x + 3) – (3x + 3) = 0
Conclusion
Since the remainder is 0, we conclude that the quotient of the division is:
x² + 2x + 3