If the polynomial `f(x) = ax^3 + bx - c` is exactly divisible by the polynomial `g(x) = x^(2) + bx + c, cne0`, then which of the following options is true?

Options

A. `c=2b^2`
B. `ab=1`
C. `ac=2b`
D. All of these

Answer

To solve the problem, we need to determine the relationship between the coefficients of the polynomial \( f(x) = ax^3 + bx - c \) and the polynomial \( g(x) = x^2 + bx + c \) given that \( f(x) \) is exactly divisible by \( g(x) \).

### Step-by-Step Solution:

1. **Understanding Divisibility**:
Since \( f(x) \) is exactly divisible by \( g(x) \), we can express this relationship as:
\[ f(x) = g(x) \cdot q(x) \]
where \( q(x) \) is some polynomial. Given the degrees of the polynomials, \( f(x) \) is of degree 3 and \( g(x) \) is of degree 2. Therefore, \( q(x) \) must be of degree 1 (since \( 2 + 1 = 3 \)).

2. **Expressing \( q(x) \)**:
Let \( q(x) = mx + n \), where \( m \) and \( n \) are constants.

3. **Expanding the Product**:
We can expand \( f(x) \) as follows:
\[ f(x) = g(x) \cdot q(x) = (x^2 + bx + c)(mx + n) \]
Expanding this gives:
\[ = mx^3 + nx^2 + bmx^2 + bnx + cmx + cn \]
Combining like terms, we get:
\[ = mx^3 + (n + bm)x^2 + (bn + cm)x + cn \]

4. **Setting Coefficients Equal**:
For \( f(x) = ax^3 + bx - c \) to hold, the coefficients of the corresponding powers of \( x \) must be equal. Thus, we can set up the following equations:
- Coefficient of \( x^3 \): \( m = a \)
- Coefficient of \( x^2 \): \( n + bm = 0 \)
- Coefficient of \( x^1 \): \( bn + cm = b \)
- Constant term: \( cn = -c \)

5. **Solving the Equations**:
From the constant term equation \( cn = -c \), since \( c \neq 0 \), we can divide both sides by \( c \):
\[ n = -1 \]

Substituting \( n = -1 \) into the second equation \( n + bm = 0 \):
\[ -1 + bm = 0 \implies bm = 1 \implies b = \frac{1}{m} \]

Now substituting \( m = a \):
\[ b = \frac{1}{a} \]

Finally, substituting \( n = -1 \) and \( m = a \) into the third equation \( bn + cm = b \):
\[ b(-1) + ca = b \implies -b + ca = b \implies ca = 2b \]

6. **Final Relationships**:
We have derived the relationships:
- \( b = \frac{1}{a} \)
- \( ca = 2b \)

### Conclusion:
From the relationships derived, we can conclude that the coefficients \( a \), \( b \), and \( c \) are related such that \( ca = 2b \) and \( b = \frac{1}{a} \).