The roots of ` 48x^3 - 44 x^2 + 12 x -1=0` are in

Options

A. A.P
B. G.P
C. H.P
D. A.G.P

Answer

To solve the equation \( 48x^3 - 44x^2 + 12x - 1 = 0 \) and find the roots, we will follow these steps:

### Step 1: Use the Rational Root Theorem
We will start by testing possible rational roots of the polynomial. A good candidate to test is \( x = \frac{1}{2} \).

### Step 2: Substitute \( x = \frac{1}{2} \) into the equation
Substituting \( x = \frac{1}{2} \):

\[ 48\left(\frac{1}{2}\right)^3 - 44\left(\frac{1}{2}\right)^2 + 12\left(\frac{1}{2}\right) - 1 \]

Calculating each term:

\[ = 48 \cdot \frac{1}{8} - 44 \cdot \frac{1}{4} + 12 \cdot \frac{1}{2} - 1 \]
\[ = 6 - 11 + 6 - 1 = 0 \]

Since the result is 0, \( x = \frac{1}{2} \) is a root.

### Step 3: Polynomial Division
Now we will divide the polynomial \( 48x^3 - 44x^2 + 12x - 1 \) by \( x - \frac{1}{2} \) using synthetic or polynomial long division.

To perform polynomial long division:

1. Multiply \( x - \frac{1}{2} \) by \( 48x^2 \) to get \( 48x^3 - 24x^2 \).
2. Subtract this from the original polynomial:

\[ (48x^3 - 44x^2 + 12x - 1) - (48x^3 - 24x^2) = -20x^2 + 12x - 1 \]

3. Next, multiply \( x - \frac{1}{2} \) by \( -20x \) to get \( -20x^2 + 10x \).
4. Subtract again:

\[ (-20x^2 + 12x - 1) - (-20x^2 + 10x) = 2x - 1 \]

5. Finally, multiply \( x - \frac{1}{2} \) by \( 2 \) to get \( 2x - 1 \).
6. Subtract:

\[ (2x - 1) - (2x - 1) = 0 \]

The result of the division is \( 48x^2 - 20x + 2 \).

### Step 4: Solve the Quadratic Equation
Now we solve the quadratic equation \( 48x^2 - 20x + 2 = 0 \) using the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Where \( a = 48, b = -20, c = 2 \):

\[ x = \frac{20 \pm \sqrt{(-20)^2 - 4 \cdot 48 \cdot 2}}{2 \cdot 48} \]
\[ = \frac{20 \pm \sqrt{400 - 384}}{96} \]
\[ = \frac{20 \pm \sqrt{16}}{96} \]
\[ = \frac{20 \pm 4}{96} \]

Calculating the two possible values:

1. \( x = \frac{24}{96} = \frac{1}{4} \)
2. \( x = \frac{16}{96} = \frac{1}{6} \)

### Step 5: List the Roots
The roots of the original polynomial are:

\[ x = \frac{1}{2}, \quad x = \frac{1}{4}, \quad x = \frac{1}{6} \]

### Step 6: Check if the Roots are in Harmonic Progression (HP)
To check if these roots are in HP, we take their reciprocals:

- \( \frac{1}{\frac{1}{2}} = 2 \)
- \( \frac{1}{\frac{1}{4}} = 4 \)
- \( \frac{1}{\frac{1}{6}} = 6 \)

The sequence \( 2, 4, 6 \) is an arithmetic progression (AP), which means that the original roots are in HP.

### Conclusion
The roots of the equation \( 48x^3 - 44x^2 + 12x - 1 = 0 \) are \( \frac{1}{2}, \frac{1}{4}, \frac{1}{6} \) and they are in HP.

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