Write the partial fraction decomposition of \( \frac{-8}{3 x^{2}-2 x-21} \)

To perform the partial fraction decomposition of the rational function: \[ \frac{-8}{3x^{2} - 2x - 21} \] **Step 1: Factor the Denominator** First, factor the quadratic denominator \(3x^{2} - 2x - 21\). 1. Multiply the coefficient of \(x^2\) (which is 3) by the constant term (-21): \[ 3 \times (-21) = -63 \] 2. Find two numbers that multiply to -63 and add up to -2. These numbers are -9 and 7. 3. Rewrite the quadratic using these numbers: \[ 3x^{2} - 9x + 7x - 21 \] 4. Factor by grouping: \[ 3x(x - 3) + 7(x - 3) = (3x + 7)(x - 3) \] So, the denominator factors as: \[ 3x^{2} - 2x - 21 = (3x + 7)(x - 3) \] **Step 2: Set Up the Partial Fractions** Express the original fraction as a sum of partial fractions: \[ \frac{-8}{(3x + 7)(x - 3)} = \frac{A}{3x + 7} + \frac{B}{x - 3} \] **Step 3: Solve for \(A\) and \(B\)** 1. Multiply both sides by the denominator \((3x + 7)(x - 3)\): \[ -8 = A(x - 3) + B(3x + 7) \] 2. Expand and collect like terms: \[ -8 = Ax - 3A + 3Bx + 7B \] \[ -8 = (A + 3B)x + (-3A + 7B) \] 3. Set up a system of equations by equating coefficients: \[ \begin{cases} A + 3B = 0 \\ -3A + 7B = -8 \end{cases} \] 4. Solve the system: - From the first equation: \(A = -3B\) - Substitute into the second equation: \[ -3(-3B) + 7B = -8 \implies 9B + 7B = -8 \implies 16B = -8 \implies B = -\frac{1}{2} \] - Then, \(A = -3(-\frac{1}{2}) = \frac{3}{2}\) **Step 4: Write the Partial Fraction Decomposition** Substitute \(A\) and \(B\) back into the partial fractions: \[ \frac{-8}{3x^{2} - 2x - 21} = \frac{\frac{3}{2}}{3x + 7} - \frac{\frac{1}{2}}{x - 3} \] Simplifying, we get: \[ \frac{-8}{3x^{2} - 2x - 21} = \frac{3}{2(3x + 7)} - \frac{1}{2(x - 3)} \] **Final Answer:** \[ \frac{-8}{3x^{2} - 2x - 21} = \frac{3}{2(3x + 7)}\, -\, \frac{1}{2(x - 3)} \]