Find the remainder when `x^(3)+3x^(2)+3x+1` is divided by :
`(i) x+1 " " (ii) x-(1)/(2) " " (iii) x " " (iv) x+pi " " (v) 5+2x`

Explanation

To find the remainder when the polynomial \( p(x) = x^3 + 3x^2 + 3x + 1 \) is divided by different linear factors, we can use the Remainder Theorem. According to this theorem, the remainder of the division of a polynomial \( p(x) \) by \( x - a \) is \( p(a) \).

Let's solve the problem step by step for each divisor.

### (i) Dividing by \( x + 1 \)

1. **Identify the value of \( a \)**: Here, \( x + 1 = 0 \) implies \( x = -1 \).
2. **Calculate \( p(-1) \)**:
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