If 1 is a twice repeated root of the equation $ax^{3}+bx^{2}+bx+d=0$, then:

Question

Accepted Answer

To determine why Option 2 ($a = d = -b$) is the correct answer, we need to analyze the given polynomial equation $ax^3 + bx^2 + bx + d = 0$ under the condition that 1 is a twice repeated root. This means that the polynomial can be expressed as $(x - 1)^2(x - r)$ for some real number $r$.

### Step-by-step Explanation:

1. **Understanding the Roots:**
   - Since 1 is a twice repeated root, the polynomial can be factored as $(x - 1)^2(x - r)$.
   - Expanding this, we get: $(x - 1)^2 = x^2 - 2x + 1$.
   - Therefore, $(x - 1)^2(x - r) = (x^2 - 2x + 1)(x - r)$.

2. **Expanding the Polynomial:**
   - Expanding $(x^2 - 2x + 1)(x - r)$ gives:
     $$
     x^3 - rx^2 - 2x^2 + 2rx + x - r
     $$
   - Simplifying, we have:
     $$
     x^3 - (r + 2)x^2 + (2r + 1)x - r
     $$

3. **Matching Coefficients:**
   - The given polynomial is $ax^3 + bx^2 + bx + d = 0$.
   - By comparing coefficients with $x^3 - (r + 2)x^2 + (2r + 1)x - r$, we get:
     - Coefficient of $x^3$: $a = 1$
     - Coefficient of $x^2$: $b = -(r + 2)$
     - Coefficient of $x$: $b = 2r + 1$
     - Constant term: $d = -r$

4. **Solving for $r$:**
   - From $b = -(r + 2)$ and $b = 2r + 1$, we equate:
     $$
     -(r + 2) = 2r + 1
     $$
   - Solving for $r$, we get:
     $$
     -r - 2 = 2r + 1 \implies -3r = 3 \implies r = -1
     $$

5. **Substituting $r = -1$:**
   - Substitute $r = -1$ into the expressions for $b$ and $d$:
     - $b = -(r + 2) = -(-1 + 2) = -1$
     - $d = -r = -(-1) = 1$

6. **Verifying Option 2:**
   - We have $a = 1$, $b = -1$, and $d = 1$.
   - This matches Option 2: $a = d = -b$.

### Explanation of Incorrect Options:

- **Option 1 ($a = d = b$):** This would imply $1 = 1 = -1$, which is not possible.
- **Option 3 ($a = -d = b$):** This would imply $1 = -1 = -1$, which contradicts $a = 1$.
- **Option 4 ($-a = d = -b$):** This would imply $-1 = 1 = 1$, which is incorrect as $a = 1$.

Thus, the correct answer is indeed Option 2, where $a = d = -b$.