Solve the equation
` 15x^3 - 23x^2 + 9x -1=0`

Answer

To solve the cubic equation \( 15x^3 - 23x^2 + 9x - 1 = 0 \), we will use the method of finding rational roots and then factor the polynomial.

### Step 1: Check for Rational Roots
We can use the Rational Root Theorem to check for possible rational roots. The possible rational roots are the factors of the constant term divided by the factors of the leading coefficient.

The constant term is \(-1\) and the leading coefficient is \(15\). The factors of \(-1\) are \(\pm 1\) and the factors of \(15\) are \(\pm 1, \pm 3, \pm 5, \pm 15\). Thus, the possible rational roots are:
\[ \pm 1, \pm \frac{1}{3}, \pm \frac{1}{5}, \pm \frac{1}{15} \]

### Step 2: Test \(x = 1\)
Let's test \(x = 1\):
\[ 15(1)^3 - 23(1)^2 + 9(1) - 1 = 15 - 23 + 9 - 1 = 0 \]
Since this equals zero, \(x = 1\) is a root.

### Step 3: Factor the Polynomial
Since \(x - 1\) is a factor, we can perform polynomial long division or synthetic division to divide \(15x^3 - 23x^2 + 9x - 1\) by \(x - 1\).

Using synthetic division:
\[ \begin{array}{r|rrrr} 1 & 15 & -23 & 9 & -1 \\ & & 15 & -8 & 1 \\ \hline & 15 & -8 & 1 & 0 \\ \end{array} \]
The result is \(15x^2 - 8x + 1\).

### Step 4: Solve the Quadratic Equation
Now we need to solve the quadratic equation \(15x^2 - 8x + 1 = 0\). We can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \(a = 15\), \(b = -8\), and \(c = 1\).

Calculating the discriminant:
\[ b^2 - 4ac = (-8)^2 - 4(15)(1) = 64 - 60 = 4 \]
Now substituting into the quadratic formula:
\[ x = \frac{8 \pm \sqrt{4}}{2 \cdot 15} = \frac{8 \pm 2}{30} \]
This gives us two solutions:
\[ x = \frac{10}{30} = \frac{1}{3} \quad \text{and} \quad x = \frac{6}{30} = \frac{1}{5} \]

### Step 5: Collect All Roots
The roots of the original cubic equation are:
\[ x = 1, \quad x = \frac{1}{3}, \quad x = \frac{1}{5} \]

### Final Answer
The solutions to the equation \(15x^3 - 23x^2 + 9x - 1 = 0\) are:
\[ x = 1, \quad x = \frac{1}{3}, \quad x = \frac{1}{5} \]