Question

17-22. Give the partial fraction decomposition for the following expressions.20. (x² - 4x + 11) / ((x - 3)(x - 1)(x + 1))

Accepted Answer

Identify the form of the partial fraction decomposition. Since the denominator is a product of three distinct linear factors $$(x - 3)$$, $$(x - 1)$$, and $$(x + 1)$$, the decomposition will be of the form:

$$\frac{A}{x - 3} + \frac{B}{x - 1} + \frac{C}{x + 1}$$ Write the equation equating the original fraction to the sum of partial fractions:

$$\frac{x^{2} - 4x + 11}{(x - 3)(x - 1)(x + 1)} = \frac{A}{x - 3} + \frac{B}{x - 1} + \frac{C}{x + 1}$$ Multiply both sides of the equation by the common denominator $$(x - 3)(x - 1)(x + 1) to $$clear the denominators:

$$x^{2} - 4x + 11 = A(x - 1)(x + 1) + B(x - 3)(x + 1) + C(x - 3)(x - 1)$$ Expand each product on the right-hand side and combine like terms to express the right side as a polynomial in $$x$$:

- Expand $$A(x - 1)(x + 1) 
- $$Expand $$B(x - 3)(x + 1) 
- $$Expand $$C(x - 3)(x - 1)$$

Then combine all terms to get a polynomial in the form $$Px^{2} + Qx + R$. Set the coefficients of corresponding powers of x$ on both sides equal to each other to form a system of equations:

- Coefficient of x$$^{2}$$: Left side equals Right side 
- Coefficient of x$: Left side equals Right side 
- Constant term: Left side equals Right side

Solve this system for A$, B$, and C$.

17-22. Give the partial fraction decomposition for the following expressions.
20. (x² - 4x + 11) / ((x - 3)(x - 1)(x + 1))

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Key Concepts

Partial Fraction Decomposition

Factoring the Denominator

Setting Up and Solving for Coefficients