Recomposition Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Recomposition.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Combining solved sub-problems back into a coherent solution for the original, larger problem.

After decomposing a problem, you must reassemble the pieces correctly — like completing a jigsaw puzzle, the boundary conditions between parts must match.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Recomposition is not automatic after decomposition — the partial answers must be combined carefully, checking that they fit together at boundaries.

Common stuck point: Don't forget to recombine—the pieces alone aren't the answer.

Sense of Study hint: Check that your pieces cover the whole and do not overlap. Then combine using the right operation (add, multiply, or union) and verify the result matches the original.

Worked Examples

Example 1

easy
After decomposing x^2 + 5x + 6 = (x+2)(x+3), recompose to solve x^2+5x+6 = 0.

Solution

  1. 1
    Decomposition gave: x^2+5x+6 = (x+2)(x+3).
  2. 2
    Set the product equal to zero: (x+2)(x+3) = 0.
  3. 3
    Recompose the solution: either x+2=0 (giving x=-2) or x+3=0 (giving x=-3).
  4. 4
    Final answer: x \in \{-2, -3\}.

Answer

x = -2 \text{ or } x = -3
Recomposition uses the pieces obtained in decomposition to assemble the final answer. The zero-product property lets us read off solutions directly from the factored form.

Example 2

medium
Partial fractions gave \frac{5x+1}{(x+1)(x-2)} = \frac{4}{3(x+1)}+\frac{11}{3(x-2)}. Recompose to verify by adding the fractions.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
You find x=3 and y=1 from solving a system. Recompose by substituting back into both equations to verify.

Example 2

medium
Given the identity \sin^2\theta = \frac{1-\cos 2\theta}{2}, recompose: use it to evaluate \int \sin^2\theta\, d\theta.
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Related Concepts

Decomposition

Background Knowledge

These ideas may be useful before you work through the harder examples.

decomposition meta