Question

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

Short 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-by-step solution

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.

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.

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.

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.

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 $\frac{4}{x-3}-\frac{1}{x+1}$