Problem 14 - College Algebra

Question

Write the partial fraction decomposition of each rational expression. $$\frac{9 x+21}{x^{2}+2 x-15}$$

Step-by-Step Solution

Answer

After following these steps, the partial fraction decomposition for the given rational expression will be obtained.

Step 1: Factor the Denominator

Start by factoring the denominator of the fraction which is a quadratic expression \(x^{2} + 2x - 15\). The factors of -15 which add up to +2 are +5 and -3. So the quadratic factors into \((x + 5)(x - 3)\).

Step 2: Set Up Partial Fractions

Next, set up the partial fraction decomposition. The rational expression \(\frac{9x + 21}{(x + 5)(x - 3)}\) can be rewritten as the sum of two simpler fractions: \(A/(x + 5) + B/(x - 3)\), where A and B are constants to be determined.

Step 3: Solve for Fraction Coefficients

Multiply the equation \(A/(x + 5) + B/(x - 3) = \frac{9x + 21}{(x + 5)(x - 3)}\) through by the denominator \((x + 5)(x - 3)\) on both sides to clear the fractions. This gives the equation \(A(x - 3) + B(x + 5) = 9x + 21\). Now, comparing coefficients on both sides of this equation, we can write two simultaneous linear equations to solve for A and B. We have A + B = 9 and -3A + 5B = 21. Solving these two equations, we can find the values of A and B.

Step 4: Express Original Fraction in Terms of Partial Fractions

Finally, substitute the determined values of A and B into the partial fractions. The original rational function can now be expressed as the sum of these two simpler fractions.