To solve the equation \( x^3 - 9x^2 + 23x - 15 = 0 \) and determine if the integer roots are in an arithmetic progression (AP), we will follow these steps:
### Step 1: Identify the coefficients
The given polynomial is:
\[
x^3 - 9x^2 + 23x - 15 = 0
\]
Here, the coefficients are:
- \( a_0 = 1 \)
- \( a_1 = -9 \)
- \( a_2 = 23 \)
- \( a_3 = -15 \)
### Step 2: List the possible rational roots
According to the Rational Root Theorem, if \( p/q \) is a rational root, then \( p \) must be a divisor of \( a_n \) (the constant term, which is -15) and \( q \) must be a divisor of \( a_0 \) (the leading coefficient, which is 1). Since \( a_0 = 1 \), all rational roots must be integers.
The divisors of -15 are:
\[
\pm 1, \pm 3, \pm 5, \pm 15
\]
### Step 3: Test the possible integer roots
We will substitute each possible integer root into the polynomial to see if it equals zero.
1. **Testing \( x = 1 \)**:
\[
1^3 - 9(1^2) + 23(1) - 15 = 1 - 9 + 23 - 15 = 0
\]
So, \( x = 1 \) is a root.
2. **Testing \( x = 3 \)**:
\[
3^3 - 9(3^2) + 23(3) - 15 = 27 - 81 + 69 - 15 = 0
\]
So, \( x = 3 \) is a root.
3. **Testing \( x = 5 \)**:
\[
5^3 - 9(5^2) + 23(5) - 15 = 125 - 225 + 115 - 15 = 0
\]
So, \( x = 5 \) is a root.
### Step 4: List the roots
The roots we found are:
\[
x = 1, 3, 5
\]
### Step 5: Check if the roots are in AP
To check if the roots are in arithmetic progression, we need to verify if the difference between consecutive terms is constant:
- The difference between the second and first root:
\[
3 - 1 = 2
\]
- The difference between the third and second root:
\[
5 - 3 = 2
\]
Since both differences are equal, the roots \( 1, 3, 5 \) are indeed in arithmetic progression.
### Conclusion
The integer roots of the equation \( x^3 - 9x^2 + 23x - 15 = 0 \) are \( 1, 3, 5 \), and they are in arithmetic progression.
---
