Remainder when dividing polynomial by another polynomial

When dividing a polynomial by another polynomial, we use long division just like we do with numbers. The goal is to find the quotient and remainder.

Dividing x^3 + 3x^2 + 3x + 1 by 2x + 5

To divide x^3 + 3x^2 + 3x + 1 by 2x + 5, we use long division as shown below:

```
x^2 - x + 3
-------------------
2x + 5 | x^3 + 3x^2 + 3x + 1
- 2x^2 - 5x
--------------
x^2 + 3x
- x^2 - 2.5x
-------------
0.5x + 1
```

Therefore, the quotient is x^2 - x + 3 and the remainder is 0.5x + 1.

Reducing the remainder modulo 5

To find the remainder when dividing by 5, we need to reduce the remainder 0.5x + 1 modulo 5.

We can do this by finding the remainder when 0.5x is divided by 5 and adding 1.

Since 0.5 is equivalent to 2.5 modulo 5, we have:

0.5x = 2.5x (mod 5)

Adding 1 to both sides, we get:

0.5x + 1 = 2.5x + 1 (mod 5)

Therefore, the remainder when x^3 + 3x^2 + 3x + 1 is divided by 2x + 5 and reduced modulo 5 is 2.5x + 1, which is equivalent to 2x + 1 modulo 5.

Final Answer: The remainder when x^3 + 3x^2 + 3x + 1 is divided by 2x + 5 and reduced modulo 5 is 2x + 1.