Write the partial fraction decomposition of the rational expression **Title: Partial Fraction Decomposition of Rational Expressions****Objective:**Learn how to perform the partial fraction decomposition of rational expressions.**Problem Statement:**Write the partial fraction decomposition of the rational expression:\[ \frac{x}{x^2 - x - 30} \]**Explanation:**1. **Identify the Denominator:** The expression in the denominator is \(x^2 - x - 30\). We need to factor this quadratic expression.2. **Factoring the Quadratic:** - Try finding two numbers that multiply to \(-30\) and add to \(-1\) (the coefficient of \(x\)). - These numbers are \(5\) and \(-6\). - Therefore, the factorization is \((x - 6)(x + 5)\).3. **Set Up Partial Fractions:** Express the original fraction as a sum of fractions with unknown coefficients: \[ \frac{x}{(x - 6)(x + 5)} = \frac{A}{x - 6} + \frac{B}{x + 5} \]4. **Solve for Coefficients A and B:** To find \(A\) and \(B\), multiply through by the original denominator to clear the fractions: \[ x = A(x + 5) + B(x - 6) \]5. **Expand and Collect Like Terms:** Solve for \(A\) and \(B\) by substituting values for \(x\) to eliminate one of the variables or by comparing coefficients.By following these steps, you'll successfully decompose the rational expression into partial fractions.

We have to write the partial fraction decomposition of the rational expression xx2-x-30.xx2-x-30 = xx2-6x+5x-30 = xxx-6+5x-6 = xx-6x+5 ..........(1) Let xx-6x+5 = Ax-6+Bx+5 .............(2) (partial fraction decomposition) ⇒ xx-6x+5 = Ax+5+Bx-6x-6x+5 ⇒ x = Ax+5+Bx-6 (cancelling the denominators) ⇒ x = Ax+5A+Bx-6B ⇒ x = A+Bx+5A-6B Comparing the coefficients on both sides, A+B=1 ⇒B=1-A 5A-6B=0Now using B=1-A in the equation 5A-6B=0, we get 5A-61-A=0 ⇒ 5A-6+6A=0 ⇒ 11A-6=0 ⇒ 11A=6 ⇒ A=611 ∴ B = 1-A = 1-611 = 11-611 = 511 Now putting A=611 and B=511 in the equation (2), we get xx-6x+5 = 611x-6+511x+5 ⇒ xx-6x+5 = 611x-6+511x+5 ..............(3)From the equations (1) and (3), xx2-x-30 = xx-6x+5 = 611x-6+511x+5 Thus, xx2-x-30 = 611(x-6)+511(x+5)