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

Question

UpStudy AI · Accepted Answer

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:**
$$
\frac{-8}{3x^{2} - 2x - 21} = \frac{3}{2(3x + 7)}\, -\, \frac{1}{2(x - 3)}
$$
$$ \frac{-8}{3x^{2} - 2x - 21} = \frac{3}{2(3x + 7)} - \frac{1}{2(x - 3)} $$

Write the partial fraction decomposition of

Solución de inteligencia artificial de Upstudy

Responder

Solución

Bonus Knowledge

preguntas relacionadas

Latest Algebra Questions