Decomposition Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

medium

Decompose the partial fraction

\dfrac{5x+1}{(x+1)(x-2)}

into the form

\dfrac{A}{x+1} + \dfrac{B}{x-2}

Solution

1
Write $\frac{5x+1}{(x+1)(x-2)} = \frac{A}{x+1} + \frac{B}{x-2}$ . Multiply both sides by $(x+1)(x-2)$ : $5x+1 = A(x-2)+B(x+1)$ .
2
Set $x=2$ : $11 = 3B$ , so $B = \frac{11}{3}$ . Set $x=-1$ : $-4 = -3A$ , so $A = \frac{4}{3}$ .
3
Result: $\frac{5x+1}{(x+1)(x-2)} = \frac{4/3}{x+1} + \frac{11/3}{x-2}$ .

Answer

\frac{5x+1}{(x+1)(x-2)} = \frac{4}{3(x+1)} + \frac{11}{3(x-2)}

Partial fraction decomposition splits a rational function into simpler fractions, each with a linear denominator. This decomposition is essential in integration and solving differential equations.

About Decomposition

The strategy of breaking a complex mathematical object or problem into simpler, independent subproblems that can be solved separately.

Learn more about Decomposition →

More Decomposition Examples

Example 1 easy

Factorise [formula] by decomposing the expression into simpler parts.

Example 3 easy

Decompose solving the system [formula] and [formula] into steps.

Example 4 medium

Decompose the proof that [formula] is irrational into logical sub-goals.

← All Decomposition Examples Decomposition Concept

Related Concepts

Recomposition