least common denominator of \( \frac{4 x}{3 x^{2}-14 x+16} \) and \( \frac{9}{3 x^{2}-11 x+8} \)

1. Factor the denominator of the first fraction: \[ 3x^2 - 14x + 16 \] To factor, look for numbers that give the roots. Using the quadratic formula: \[ x = \frac{14 \pm \sqrt{(-14)^2 - 4 \cdot 3 \cdot 16}}{2 \cdot 3} = \frac{14 \pm \sqrt{196 - 192}}{6} = \frac{14 \pm 2}{6} \] Thus, the roots are: \[ x = \frac{16}{6} = \frac{8}{3} \quad \text{and} \quad x = \frac{12}{6} = 2 \] Therefore, the factorization is: \[ 3x^2 - 14x + 16 = (3x-8)(x-2) \] 2. Factor the denominator of the second fraction: \[ 3x^2 - 11x + 8 \] Again, using the quadratic formula: \[ x = \frac{11 \pm \sqrt{(-11)^2 - 4 \cdot 3 \cdot 8}}{2 \cdot 3} = \frac{11 \pm \sqrt{121 - 96}}{6} = \frac{11 \pm 5}{6} \] The roots obtained are: \[ x = \frac{16}{6} = \frac{8}{3} \quad \text{and} \quad x = \frac{6}{6} = 1 \] Thus, the factorization is: \[ 3x^2 - 11x + 8 = (3x-8)(x-1) \] 3. Identify the unique factors from both denominators: - From \( \frac{4x}{(3x-8)(x-2)} \): factors are \( 3x-8 \) and \( x-2 \). - From \( \frac{9}{(3x-8)(x-1)} \): factors are \( 3x-8 \) and \( x-1 \). 4. The least common denominator (LCD) is the product of the combined unique factors: \[ \text{LCD} = (3x-8)(x-2)(x-1) \]