If A, G & H are respectively the A.M., G.M. & H.M. of three positive numbers a, b, & c, then equation whose roots are a, b, & c is given by

Options

A. `x^(3) - 3A x^(2)+(3G^(3))/(H)x-G^(3)=0`
B. `x^(3) + 3A x^(2)+(3G^(3))/(H)x-G^(3)=0`
C. `x^(3) + A x^(2)+(G^(3))/(H)-G^(3)=0`
D. `x^(3) - 3A x^(2)-(3G^(3))/(H)x-G^(3)=0`

Explanation

To find the equation whose roots are the three positive numbers \( a, b, \) and \( c \), given that \( A, G, \) and \( H \) are respectively the Arithmetic Mean (A.M.), Geometric Mean (G.M.), and Harmonic Mean (H.M.) of these numbers, we will follow these steps:

### Step 1: Define the A.M., G.M., and H.M.
- The Arithmetic Mean \( A \) of the numbers \( a, b, c \) is given by:
\[ A = \frac{a + b + c}{3} \]
- The Geometric Mean \( G \) is given by:
\[ G = \sqrt[3]{abc} \]
- The Harmonic Mean \( H \) is defined as:
\[ H = \frac{3}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}} = \frac{3abc}{ab + ac + bc} \]

### Step 2: Express the relationships
From the definition of A.M., we can express:
\[ a + b + c = 3A \]
From the definition of G.M., we can express:
\[ abc = G^3 \]
From the definition of H.M., we can express:
\[ \frac{3}{H} = \frac{ab + ac + bc}{abc} \implies ab + ac + bc = \frac{3abc}{H} \]

### Step 3: Substitute the values
Now, we can substitute the values we have:
1. From \( a + b + c = 3A \)
2. From \( abc = G^3 \)
3. From \( ab + ac + bc = \frac{3G^3}{H} \)

### Step 4: Form the polynomial
The polynomial whose roots are \( a, b, c \) can be written in the form:
\[ x^3 - S_1 x^2 + S_2 x - S_3 = 0 \]
where:
- \( S_1 = a + b + c = 3A \)
- \( S_2 = ab + ac + bc = \frac{3G^3}{H} \)
- \( S_3 = abc = G^3 \)

### Step 5: Write the final equation
Substituting the values of \( S_1, S_2, \) and \( S_3 \) into the polynomial gives:
\[ x^3 - 3Ax^2 + \frac{3G^3}{H} x - G^3 = 0 \]

Thus, the equation whose roots are \( a, b, c \) is:
\[ x^3 - 3Ax^2 + \frac{3G^3}{H} x - G^3 = 0 \]