Problem 2 - College Algebra and Calculus: An Applied Approach

Question

Write the partial fraction decomposition for the expression. $$ \frac{3 x+11}{x^{2}-2 x-3} $$

Step-by-Step Solution

Answer

The partial fraction decomposition of the expression \(\frac{3x+11}{x^{2}-2x-3}\) is \(\frac{4}{x-3} - \frac{1}{x+1}\)

Step 1: Factor the Denominator

Factoring the denominator: \(x^{2}-2x-3\) = (x-3)(x+1). This is accomplished by finding two numbers that multiply to -3 and add to -2, which are -3 and 1. This gives us two denominators for our partial fractions: x-3 and x+1.

Step 2: Set up the Partial Fractions

Using the factors found in Step 1, we set up the partial fractions. This means the original expression \(\frac{3x+11}{x^{2}-2x-3}\) can be written as \(\frac{A}{x-3}+ \frac{B}{x+1}\) where A and B are numbers we need to find, representing the numerators of the partial fractions.

Step 3: Clear the fractions to find A and B

Multiply through by the common denominator (x-3)(x+1) to clear the fractions. Setting this equal to \(3x + 11\), gives us an equation \(A(x+1) + B(x - 3) = 3x + 11\). This is the polynomial equation we must solve for A and B.

Step 4: Solve for A and B

Expand \(A(x+1) + B(x-3)\) to get \(Ax + A + Bx - 3B\). Equate the coefficients in this expression with those in \(3x + 11\), we obtain two simple linear equations in A and B: \(A + B = 3\) and \(A - 3B = 11\). Solving these two equations, we find A = 4 and B = -1.

Step 5: Substitute Back to the Partial Fractions

Substitute A and B back to the partial fractions. Thus, the partial fraction decomposition of the expression \(\frac{3x+11}{x^{2}-2x-3}\) is \(\frac{4}{x-3} - \frac{1}{x+1}\)