To find the quotient of the polynomial x3 + 3x2 + 5x + 3, we can use polynomial long division to divide it by a linear term, if that’s what you meant. However, your question seems to only present a single polynomial, so it’s a bit unclear what you’re dividing by.
If we assume you want to divide this cubic polynomial by something like x + 1, here’s how to do it:
Step 1: Set up the division
Write x3 + 3x2 + 5x + 3 under the long division symbol and x + 1 outside.
Step 2: Divide the leading terms
The leading term of x3 divided by the leading term of x results in x2. We write x2 above the long division symbol.
Step 3: Multiply and subtract
Multiply x2 by x + 1, giving x3 + x2. Subtract this from the original polynomial:
(x3 + 3x2 + 5x + 3) – (x3 + x2) = 2x2 + 5x + 3
Step 4: Repeat the process
Now, repeat the division using 2x2 + 5x + 3 and x + 1:
The leading term 2x2 divided by x gives 2x. Multiply and subtract again:
(2x2 + 5x + 3) – (2x2 + 2x) = 3x + 3
Step 5: Last division
For 3x + 3, divide by x + 1 to get 3. After multiplying and subtracting, you’ll find:
0 remainder.
Conclusion
The final quotient when dividing x3 + 3x2 + 5x + 3 by x + 1 is x2 + 2x + 3
If your original question intended a different divisor, please clarify, and I’d be happy to correct the calculation!