Decomposition Math Example 1

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Example 1

easy
Factorise 12x2+8x−412x^2 + 8x - 4 by decomposing the expression into simpler parts.

Solution

  1. 1
    Step 1 — Factor out GCF: gcd⁡(12,8,4)=4\gcd(12, 8, 4) = 4. So 12x2+8x−4=4(3x2+2x−1)12x^2+8x-4 = 4(3x^2+2x-1).
  2. 2
    Step 2 — Decompose the quadratic: find two numbers multiplying to 3×(−1)=−33 \times (-1) = -3 and adding to 22: those are 33 and −1-1.
  3. 3
    Rewrite middle term: 4(3x2+3x−x−1)=4[3x(x+1)−1(x+1)]=4(3x−1)(x+1)4(3x^2 + 3x - x - 1) = 4[3x(x+1) - 1(x+1)] = 4(3x-1)(x+1).

Answer

12x2+8x−4=4(3x−1)(x+1)12x^2+8x-4 = 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.

About Decomposition

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

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More Decomposition Examples

Example 2 medium

Decompose the partial fraction [formula] into the form [formula].

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.

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Related Concepts

Recomposition