If `x^2+px+1` is a factor of `ax^3+bx+c` then a) `a^2+c^2=-ab` b) `a^2+c^2ab` c) `a^2-c^2=ab` d) `a^2-c^2=-ab`

Options

A. A) `a ^(2)+ xc ^(2) =-ab`
B. B) `a ^(2) + c ^(2) =ab`
C. C) `a ^(2) - c ^(2)=ab`
D. D) `a ^(2) - c ^(2) =-ab`

Answer

To solve the problem, we need to determine the conditions under which the polynomial \(x^2 + px + 1\) is a factor of the polynomial \(ax^3 + bx + c\).

### Step-by-step Solution:

1. **Understanding the Problem**:
We know that if \(x^2 + px + 1\) is a factor of \(ax^3 + bx + c\), then when we divide \(ax^3 + bx + c\) by \(x^2 + px + 1\), the remainder must be zero.

2. **Performing Polynomial Long Division**:
- We will divide \(ax^3 + bx + c\) by \(x^2 + px + 1\).
- The first term of the quotient will be \(a\) because \(a \cdot x^2 = ax^3\).
- Multiply \(x^2 + px + 1\) by \(a\):
\[ a(x^2 + px + 1) = ax^3 + apx^2 + a \]
- Subtract this from \(ax^3 + bx + c\):
\[ (ax^3 + bx + c) - (ax^3 + apx^2 + a) = -apx^2 + (b - a)x + (c - a) \]

3. **Continuing the Division**:
- Next, we take the term \(-apx^2\) and divide it by \(x^2\) to get \(-pa\).
- Multiply \(x^2 + px + 1\) by \(-pa\):
\[ -pa(x^2 + px + 1) = -pax^2 - p^2ax - pa \]
- Subtract this from the previous remainder:
\[ (-apx^2 + (b - a)x + (c - a)) - (-pax^2 - p^2ax - pa) = (p^2a + b - a)x + (c - a + pa) \]

4. **Setting the Remainder to Zero**:
- For \(x^2 + px + 1\) to be a factor, both coefficients of \(x\) and the constant term must equal zero:
\[ p^2a + b - a = 0 \quad \text{(1)} \]
\[ c - a + pa = 0 \quad \text{(2)} \]

5. **Solving for \(p\)**:
- From equation (2), we can express \(p\):
\[ pa = a - c \implies p = \frac{a - c}{a} \]

6. **Substituting \(p\) into Equation (1)**:
- Substitute \(p\) into equation (1):
\[ \left(\frac{a - c}{a}\right)^2 a + b - a = 0 \]
\[ \frac{(a - c)^2}{a} + b - a = 0 \]
\[ (a - c)^2 + ab - a^2 = 0 \]
\[ a^2 - 2ac + c^2 + ab - a^2 = 0 \]
\[ -2ac + c^2 + ab = 0 \]

7. **Rearranging the Equation**:
- Rearranging gives us:
\[ c^2 - 2ac + ab = 0 \]
- This can be rewritten as:
\[ a^2 - c^2 = -ab \]

### Final Answer:
The correct option is:
**d) \(a^2 - c^2 = -ab\)**.