The roots of `x^3 - 13x^2 +39 x-27=0` are in

Options

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

Answer

To determine the nature of the roots of the polynomial equation \( x^3 - 13x^2 + 39x - 27 = 0 \), we will follow these steps:

### Step 1: Identify the polynomial and check for rational roots
The given polynomial is:
\[ x^3 - 13x^2 + 39x - 27 = 0 \]
We will use the Rational Root Theorem to test for possible rational roots. The possible rational roots are the factors of the constant term (-27) divided by the factors of the leading coefficient (1).

### Step 2: Test for possible roots
Let's test \( x = 1 \):
\[ 1^3 - 13(1^2) + 39(1) - 27 = 1 - 13 + 39 - 27 = 0 \]
Since \( x = 1 \) is a root, we can factor the polynomial by \( x - 1 \).

### Step 3: Polynomial long division
Now we will divide the polynomial \( x^3 - 13x^2 + 39x - 27 \) by \( x - 1 \).

1. Divide the leading term: \( x^3 \div x = x^2 \)
2. Multiply \( x^2 \) by \( x - 1 \): \( x^3 - x^2 \)
3. Subtract:
\[ (x^3 - 13x^2) - (x^3 - x^2) = -12x^2 \]
4. Bring down the next term:
\[ -12x^2 + 39x \]
5. Divide: \( -12x^2 \div x = -12x \)
6. Multiply: \( -12x(x - 1) = -12x^2 + 12x \)
7. Subtract:
\[ (-12x^2 + 39x) - (-12x^2 + 12x) = 27x \]
8. Bring down the last term:
\[ 27x - 27 \]
9. Divide: \( 27x \div x = 27 \)
10. Multiply: \( 27(x - 1) = 27x - 27 \)
11. Subtract:
\[ (27x - 27) - (27x - 27) = 0 \]

The result of the division is:
\[ x^2 - 12x + 27 \]

### Step 4: Solve the quadratic equation
Now we need to solve the quadratic equation \( x^2 - 12x + 27 = 0 \) using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1, b = -12, c = 27 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-12)^2 - 4(1)(27) = 144 - 108 = 36 \]
Now substituting into the quadratic formula:
\[ x = \frac{12 \pm \sqrt{36}}{2} = \frac{12 \pm 6}{2} \]
This gives us:
\[ x = \frac{18}{2} = 9 \quad \text{and} \quad x = \frac{6}{2} = 3 \]

### Step 5: List all roots
The roots of the polynomial are:
\[ x = 1, \quad x = 3, \quad x = 9 \]

### Step 6: Determine the nature of the roots
To check if these roots are in an arithmetic progression (AP), geometric progression (GP), or harmonic progression (HP):

1. **AP**: The difference between consecutive terms should be constant.
- Differences: \( 3 - 1 = 2 \) and \( 9 - 3 = 6 \) (not constant, so not in AP).

2. **GP**: The ratio between consecutive terms should be constant.
- Ratios: \( \frac{3}{1} = 3 \) and \( \frac{9}{3} = 3 \) (constant, so they are in GP).

3. **HP**: The reciprocals of the roots should be in AP.
- Reciprocals: \( 1, \frac{1}{3}, \frac{1}{9} \)
- Differences: \( \frac{1}{3} - 1 = -\frac{2}{3} \) and \( \frac{1}{9} - \frac{1}{3} = -\frac{2}{9} \) (not constant, so not in HP).

### Conclusion
The roots \( 1, 3, 9 \) are in a **Geometric Progression (GP)**.