Decomposition Examples in Math: 47 Problems with Answers
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 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 12x2+8x−4 by decomposing the expression into simpler parts.
Answer
12x2+8x−4=4(3x−1)(x+1)
First step
1
Step 1 — Factor out GCF: gcd(12,8,4)=4. So 12x2+8x−4=4(3x2+2x−1).
Full solution
2
Step 2 — Decompose the quadratic: find two numbers multiplying to 3×(−1)=−3 and adding to 2: those are 3 and −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 (x+1)(x−2)5x+1 into the form x+1A+x−2B.
Example 3
medium
Decompose x2+x2x+1 into partial fractions.
Example 4
medium
Decompose cos(2θ) into a function of cosθ only.
Example 5
hard
Decompose ∫x2−11dx 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 2+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 43 as a sum of unit-related parts to compute 43 of 20.
Example 5
easy
To compute 7×12, 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 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 x2−15x+1 into partial fractions (state the form and find constants).
Example 12
medium
Decompose ∫(2x+cosx)dx into manageable subintegrals.
Example 13
medium
Decompose the area between y=x and y=x2 on [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 n3−n is divisible by 6 for all integers n.
Example 17
challenge
Decompose ∑k=1nk(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 127 into a sum of two unit-fraction-friendly parts to add to 121.
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 4×5 rectangle on top of a 6×2 rectangle, what is a natural decomposition?
Example 25
easy
Decompose 65 as a sum of unit fractions.
Example 26
easy
Decompose 1024 as a power of 2.
Example 27
easy
To compute 99×7, decompose the 99.
Example 28
easy
Decompose 50 to simplify it.
Example 29
medium
Decompose the count of integers from 1 to 100 divisible by 2 or 3 using inclusion-exclusion.
Example 30
medium
Decompose the integral ∫(3x2+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 x3−8 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 ∑k=1n(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=sinx and the x-axis from 0 to π.
Example 39
hard
Decompose the count of ways to arrange the letters of MISSISSIPPI.
Example 40
hard
Decompose a 7×8 chessboard count of dominoes (1×2 tiles). Approximate using a known result.
Example 41
challenge
Decompose the proof that any integer n≥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,…,n} with no two consecutive elements (Fibonacci connection).