Find the values of `a,b,c,d,` if 1,2,3,4 are the roots of `x^4 +ax^3 +bx^2 +cx +d=0`

Answer

To find the values of \( a, b, c, d \) in the polynomial equation \( x^4 + ax^3 + bx^2 + cx + d = 0 \) given that the roots are \( 1, 2, 3, 4 \), we can follow these steps:

### Step 1: Write the polynomial in factored form
Since the roots of the polynomial are \( 1, 2, 3, \) and \( 4 \), we can express the polynomial as:
\[ (x - 1)(x - 2)(x - 3)(x - 4) = 0 \]

### Step 2: Expand the factors
We can first multiply the factors in pairs.

1. Multiply the first two factors:
\[ (x - 1)(x - 2) = x^2 - 3x + 2 \]

2. Multiply the last two factors:
\[ (x - 3)(x - 4) = x^2 - 7x + 12 \]

### Step 3: Multiply the results from Step 2
Now we multiply the two quadratic results:
\[ (x^2 - 3x + 2)(x^2 - 7x + 12) \]

Using the distributive property (also known as the FOIL method for binomials):
\[ = x^2(x^2 - 7x + 12) - 3x(x^2 - 7x + 12) + 2(x^2 - 7x + 12) \]

Expanding each term:
1. \( x^2(x^2 - 7x + 12) = x^4 - 7x^3 + 12x^2 \)
2. \( -3x(x^2 - 7x + 12) = -3x^3 + 21x^2 - 36x \)
3. \( 2(x^2 - 7x + 12) = 2x^2 - 14x + 24 \)

### Step 4: Combine like terms
Now we combine all the terms:
\[ x^4 + (-7x^3 - 3x^3) + (12x^2 + 21x^2 + 2x^2) + (-36x - 14x) + 24 \]
This simplifies to:
\[ x^4 - 10x^3 + 35x^2 - 50x + 24 \]

### Step 5: Identify coefficients
Now we can compare this polynomial with the standard form \( x^4 + ax^3 + bx^2 + cx + d \):
- Coefficient of \( x^3 \): \( a = -10 \)
- Coefficient of \( x^2 \): \( b = 35 \)
- Coefficient of \( x \): \( c = -50 \)
- Constant term: \( d = 24 \)

### Final Values
Thus, the values of \( a, b, c, d \) are:
\[ a = -10, \quad b = 35, \quad c = -50, \quad d = 24 \]

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