If 1,2,3 and 4 are the roots of the equation `x^4 + ax^3 + bx^2 +cx +d=0` then ` a+ 2b +c=`

Options

A. `-25`
B. 0
C. 10
D. 24

Answer

To solve the problem, we need to find the values of \( a \), \( b \), and \( c \) in the polynomial equation \( x^4 + ax^3 + bx^2 + cx + d = 0 \), given that the roots of the equation are 1, 2, 3, and 4.

### Step-by-Step Solution:

1. **Write the polynomial using its roots**:
Since the roots of the polynomial are 1, 2, 3, and 4, we can express the polynomial as:
\[ (x - 1)(x - 2)(x - 3)(x - 4) = 0 \]

2. **Expand the polynomial**:
We will expand the product step by step:
- First, multiply the first two factors:
\[ (x - 1)(x - 2) = x^2 - 3x + 2 \]
- Next, multiply the last two factors:
\[ (x - 3)(x - 4) = x^2 - 7x + 12 \]
- Now, multiply the two quadratic results:
\[ (x^2 - 3x + 2)(x^2 - 7x + 12) \]

3. **Distributing the multiplication**:
- Multiply \( x^2 \) with each term in the second polynomial:
\[ x^2 \cdot (x^2 - 7x + 12) = x^4 - 7x^3 + 12x^2 \]
- Multiply \( -3x \) with each term in the second polynomial:
\[ -3x \cdot (x^2 - 7x + 12) = -3x^3 + 21x^2 - 36x \]
- Multiply \( 2 \) with each term in the second polynomial:
\[ 2 \cdot (x^2 - 7x + 12) = 2x^2 - 14x + 24 \]

4. **Combine all the terms**:
Now, combine all the results:
\[ x^4 + (-7x^3 - 3x^3) + (12x^2 + 21x^2 + 2x^2) + (-36x - 14x) + 24 \]
This simplifies to:
\[ x^4 - 10x^3 + 35x^2 - 50x + 24 \]

5. **Identify coefficients**:
From the expanded polynomial, we can identify the coefficients:
- \( a = -10 \)
- \( b = 35 \)
- \( c = -50 \)
- \( d = 24 \)

6. **Calculate \( a + 2b + c \)**:
Now, substitute the values of \( a \), \( b \), and \( c \) into the expression:
\[ a + 2b + c = -10 + 2(35) - 50 \]
Simplifying this gives:
\[ -10 + 70 - 50 = 10 \]

### Final Answer:
Thus, the value of \( a + 2b + c \) is \( \boxed{10} \).