Decomposition Examples in Math

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

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

Concept Recap

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

Divide and conquer: a hard problem of size $n$ becomes $n$ easy problems. Long division, partial fractions, and integration by parts all use decomposition.

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: Decomposition splits a hard problem into simpler, independent subproblems you can solve separately, then handle on their own.

Common stuck point: The procedure for decomposition is the easy part; the trap is splitting into parts that secretly depend on each other. Asking "Can I split this into simpler subproblems that each be solved on their own?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Can I split this into simpler subproblems that each be solved on their own?

Worked Examples

Example 1

easy

Factorise

12x^2 + 8x - 4

by decomposing the expression into simpler parts.

Answer

12x^2+8x-4 = 4(3x-1)(x+1)

First step

Step 1 — Factor out GCF:

\gcd(12, 8, 4) = 4

. So

12x^2+8x-4 = 4(3x^2+2x-1)

Full solution

2
Step 2 — Decompose the quadratic: find two numbers multiplying to $3 \times (-1) = -3$ and adding to $2$ : those are $3$ and $-1$ .
3
Rewrite middle term: $4(3x^2 + 3x - x - 1) = 4[3x(x+1) - 1(x+1)] = 4(3x-1)(x+1)$ .

Decomposition breaks a complex factoring problem into stages: first extract the GCF, then factor the remaining quadratic by splitting the middle term. Each stage is simpler than the whole.

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}

Example 3

medium

Decompose

\frac{2x+1}{x^2+x}

into partial fractions.

Example 4

medium

Decompose

\cos(2\theta)

into a function of

\cos\theta

only.

Example 5

hard

Decompose

\int \frac{1}{x^2-1}\,dx

via partial fractions and evaluate.

Practice Problems

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

Example 1

easy

Decompose solving the system

2x + y = 7

and

x - y = 2

into steps.

Example 2

medium

Decompose the proof that

\sqrt{2} + \sqrt{3}

is irrational into logical sub-goals.

Example 3

easy

To find the area of an L-shaped region, what is a natural decomposition?

Example 4

easy

Decompose

\frac{3}{4}

as a sum of unit-related parts to compute

\frac{3}{4}

20

Example 5

easy

To compute

7 \times 12

, decompose

12

into a sum. What is a useful split?

Example 6

easy

Decompose the problem 'find the perimeter of a rectangle' into subparts.

Example 7

easy

Decompose

\sqrt{72}

to simplify it.

Example 8

easy

Decompose the count: how many ways to pick an outfit from

3

shirts and

4

pants?

Example 9

easy

Decompose

48

into prime factors.

Example 10

easy

Decompose the journey: a trip is

40

km at

20

km/h then

30

km at

30

km/h. Into what subproblems?

Example 11

medium

Decompose

\frac{5x+1}{x^2-1}

into partial fractions (state the form and find constants).

Example 12

medium

Decompose

\int (2x + \cos x)\,dx

into manageable subintegrals.

Example 13

medium

Decompose the area between

y=x

and

y=x^2

[0,1]

into the right subproblem.

Example 14

medium

Decompose

123

in base

10

to show place-value structure.

Example 15

medium

Decompose the counting of arrangements of the letters in 'BOOK' into stages.

Example 16

challenge

Decompose the proof that

n^3-n

is divisible by

6

for all integers

n

Example 17

challenge

Decompose

\sum_{k=1}^{n} k(k+1)

into simpler known sums and evaluate for general

n

Example 18

challenge

Decompose the volume of a cylinder with a cone removed from its top (same radius

r

, both height

h

Example 19

medium

Decompose the count of

3

-digit numbers with distinct digits into stages.

Example 20

medium

Decompose

\frac{7}{12}

into a sum of two unit-fraction-friendly parts to add to

\frac{1}{12}

Example 21

medium

Decompose the area of a trapezoid (parallel sides

a,b

, height

h

) into a rectangle plus triangles to derive its formula.

Example 22

medium

Decompose solving

|x-3|=5

into cases.

Example 23

easy

Decompose

98

to compute

98 + 47

mentally.

Example 24

easy

To find the area of a shape made of a

4\times 5

rectangle on top of a

6\times 2

rectangle, what is a natural decomposition?

Example 25

easy

Decompose

\frac{5}{6}

as a sum of unit fractions.

Example 26

easy

Decompose

1024

as a power of

2

Example 27

easy

To compute

99 \times 7

, decompose the

99

Example 28

easy

Decompose

\sqrt{50}

to simplify it.

Example 29

medium

Decompose the count of integers from

1

100

divisible by

2

3

using inclusion-exclusion.

Example 30

medium

Decompose the integral

\int (3x^2 + 2x + 5)\,dx

into simpler integrals.

Example 31

medium

Decompose the area of a regular hexagon (side

s

) into triangles.

Example 32

medium

Decompose

108

into prime factors and use it to find the number of divisors.

Example 33

medium

Decompose

x^3 - 8

as a product of factors.

Example 34

medium

Decompose the count of

4

-letter words from

\{A,B,C\}

that contain at least one

A

Example 35

medium

Decompose the volume of a cone (radius

r

, height

h

) as a fraction of a cylinder.

Example 36

medium

Decompose

\sum_{k=1}^{n} (2k-1)

into a known sum and evaluate.

Example 37

hard

Decompose the proof that the product of two odd numbers is odd.

Example 38

hard

Decompose the area between

y=\sin x

and the

x

-axis from

0

\pi

Example 39

hard

Decompose the count of ways to arrange the letters of MISSISSIPPI.

Example 40

hard

Decompose a

7 \times 8

chessboard count of dominoes (

1\times 2

tiles). Approximate using a known result.

Example 41

challenge

Decompose the proof that any integer

n \ge 2

can be written as a product of primes (existence half of the Fundamental Theorem of Arithmetic).

Example 42

challenge

Decompose the count of subsets of

\{1,2,\dots,n\}

with no two consecutive elements (Fibonacci connection).

← Back to Decomposition Practice Mode

Related Concepts

Recomposition