If the roots of `x^(4) - 2x^(3) - 21x^(2) + 22x + 40 = 0 ` are in A.P. then the roots are

Answer

To find the roots of the polynomial \( x^4 - 2x^3 - 21x^2 + 22x + 40 = 0 \) given that the roots are in Arithmetic Progression (A.P.), we can follow these steps:

### Step 1: Assume the Roots
Since the roots are in A.P., we can denote them as:
- \( a - 3d \)
- \( a - d \)
- \( a + d \)
- \( a + 3d \)

### Step 2: Use Vieta's Formulas
According to Vieta's formulas, the sum of the roots (denoted as \( S_1 \)) can be expressed as:
\[ S_1 = (a - 3d) + (a - d) + (a + d) + (a + 3d) = 4a \]
From the polynomial, the sum of the roots is given by \( -\frac{b}{a} = -\frac{-2}{1} = 2 \). Therefore, we have:
\[ 4a = 2 \implies a = \frac{1}{2} \]

### Step 3: Calculate the Product of the Roots
The product of the roots (denoted as \( S_4 \)) can be expressed as:
\[ S_4 = (a - 3d)(a - d)(a + d)(a + 3d) \]
This can be simplified using the identity for the product of two pairs:
\[ S_4 = [(a^2 - 9d^2)(a^2 - d^2)] \]
From Vieta's, the product of the roots is given by \( \frac{e}{a} = \frac{40}{1} = 40 \). Thus, we have:
\[ (a^2 - 9d^2)(a^2 - d^2) = 40 \]

### Step 4: Substitute \( a \)
Substituting \( a = \frac{1}{2} \) into the equation:
\[ \left(\left(\frac{1}{2}\right)^2 - 9d^2\right)\left(\left(\frac{1}{2}\right)^2 - d^2\right) = 40 \]
This simplifies to:
\[ \left(\frac{1}{4} - 9d^2\right)\left(\frac{1}{4} - d^2\right) = 40 \]

### Step 5: Expand and Rearrange
Expanding the left-hand side:
\[ \left(\frac{1}{4} - 9d^2\right)\left(\frac{1}{4} - d^2\right) = \frac{1}{16} - \frac{1}{4}d^2 - \frac{9}{4}d^2 + 9d^4 = 40 \]
This leads to:
\[ 9d^4 - \frac{10}{4}d^2 + \frac{1}{16} - 40 = 0 \]
Multiplying through by 16 to eliminate the fraction:
\[ 144d^4 - 40d^2 - 639 = 0 \]

### Step 6: Let \( y = d^2 \)
Let \( y = d^2 \):
\[ 144y^2 - 40y - 639 = 0 \]

### Step 7: Solve the Quadratic Equation
Using the quadratic formula:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{40 \pm \sqrt{(-40)^2 - 4 \times 144 \times (-639)}}{2 \times 144} \]
Calculating the discriminant:
\[ = \frac{40 \pm \sqrt{1600 + 4 \times 144 \times 639}}{288} \]
After solving, we find the values of \( y \).

### Step 8: Find \( d \) and Roots
Once \( y \) is found, take the square root to find \( d \). Substitute back to find the roots:
- \( a - 3d \)
- \( a - d \)
- \( a + d \)
- \( a + 3d \)

### Final Roots
After calculating, we find the roots to be:
- \( -4, -1, 2, 5 \)