In each of the following polynomials, find the value of a if `x+a` is a factor of
(i) `x^(3)+ax^(2)-2x+a+4`
(ii) `x^(4)-a^(2)x^(2)+3x-a`

Answer

To solve the problem, we need to find the value of \( a \) such that \( x + a \) is a factor of the given polynomials. We will use the Remainder Theorem, which states that if \( x + a \) is a factor of a polynomial \( f(x) \), then \( f(-a) = 0 \).

### Part (i): Polynomial \( f(x) = x^3 + ax^2 - 2x + a + 4 \)

1. **Set up the equation**: Since \( x + a \) is a factor, we set \( x = -a \).

\[ f(-a) = (-a)^3 + a(-a)^2 - 2(-a) + a + 4 \]

2. **Substitute and simplify**:
\[ f(-a) = -a^3 + a(a^2) + 2a + a + 4 \]
\[ = -a^3 + a^3 + 2a + a + 4 \]
\[ = 3a + 4 \]

3. **Set the equation to zero**:
\[ 3a + 4 = 0 \]

4. **Solve for \( a \)**:
\[ 3a = -4 \implies a = -\frac{4}{3} \]

### Part (ii): Polynomial \( g(x) = x^4 - a^2x^2 + 3x - a \)

1. **Set up the equation**: Again, since \( x + a \) is a factor, we set \( x = -a \).

\[ g(-a) = (-a)^4 - a^2(-a)^2 + 3(-a) - a \]

2. **Substitute and simplify**:
\[ g(-a) = a^4 - a^2(a^2) - 3a - a \]
\[ = a^4 - a^4 - 4a \]
\[ = -4a \]

3. **Set the equation to zero**:
\[ -4a = 0 \]

4. **Solve for \( a \)**:
\[ a = 0 \]

### Final Answers:
- For part (i), \( a = -\frac{4}{3} \)
- For part (ii), \( a = 0 \)