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If multiplying out brackets is like unpacking a box, **factorising** is packing it back up into its neatest, most organized form.

Every algebraic term has a **coefficient** (the number multiplying the variable, like $3$ in $3x$) and variable parts, and factorising lets us uncover what they all share.

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<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
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        factor: '3',
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How do you figure out exactly what number or variable belongs on the outside of the brackets?

To factorise completely, we find the **highest common factor** (HCF) — the largest value that divides every term with no remainder.

In $12x + 18$, the numbers $12$ and $18$ share common factors of $1$, $2$, $3$, and $6$, but $6$ is the largest.

<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<meta name="aspect-ratio" content="4:3">
<title>Factorising Common Factors</title>
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css">
<script src="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.js"></script>
<style>
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</head>
<body>

<div class="main-area">
  
  <div class="step-nav">
    <button class="step-btn active" onclick="setStep(0)">1. Terms</button>
    <button class="step-btn" onclick="setStep(1)">2. Find HCF</button>
    <button class="step-btn" onclick="setStep(2)">3. Factorised</button>
  </div>

  
  <div class="math-display" id="mathOutput"></div>

  
  <div class="visual-container">
    <div class="area-model">
      
      <div class="side-factor" id="sideFactor">6</div>

      
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        <div class="block block-18">
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  <div class="controls">
    <button class="play-btn" id="playBtn" onclick="togglePlay()" title="Play / Pause">⏸</button>
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</div>

<script>
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  ];

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    }
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  // Initialize
  setStep(0);
  startTimer();
</script>

</body>
</html>

Numbers are straightforward, but what happens when the terms share letters with powers?

When terms share the same variable, look for the lowest exponent present across all terms — that lowest power is your variable factor.

For $4x^3 + 6x^2$, the numerical HCF is $2$ and the variable HCF is $x^2$, giving a total HCF of $2x^2$ to pull outside: $2x^2(2x + 3)$.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <title>Factorising by Common Factor</title>
  <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.css">
  <script src="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.js"></script>
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    }
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      background: #ffffff;
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      border-radius: 8px;
      padding: 6px 12px;
      font-size: 13px;
      font-weight: 600;
      color: #64748b;
      cursor: pointer;
      box-shadow: 0 1px 2px rgba(0,0,0,0.05);
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      border-radius: 8px;
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      cursor: pointer;
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      max-width: 480px;
      background: #ffffff;
      border-radius: 12px;
      border: 1px solid #e2e8f0;
      padding: 16px 12px;
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What happens if an entire term divides out completely and seems to leave nothing behind?

When a term matches the HCF exactly, dividing it leaves $1$, never $0$ — so $5x + 5$ factorises to $5(x + 1)$ because $5 \div 5 = 1$.

If the leading term is negative, such as $-8a - 12b$, factoring out $-4$ flips both signs inside the bracket, resulting in $-4(2a + 3b)$.

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Whenever you finish factorising, quickly expand your brackets mentally — if multiplying back produces your original expression, your factorisation is fully correct.

math

Factorising by Taking Out a Common Factor. Factorise algebraic expressions by identifying and removing the highest common factor.

Factoring and the Distributive Property / Taking Common Factors

Factorising Single Brackets - Algebraic Expressions

Year 9 Maths: Factorise Algebraic Expressions using the Highest Common Factor

Teaching factorisation by taking out a common factor in Years 9–10 Australian Curriculum (v9) serves as the crucial bridge between expanding linear brackets and factorising monic quadratics (AC9M9A02). Effective pedagogical practice starts with concrete/visual models (area grids and algebra tiles) before moving to abstract division. Emphasise writing out expanded prime factors for struggling students and explicitly train students to check their solutions by expansion to catch zero-remainder and sign errors.

Factorising by Taking Out a Common Factor

Finding the Highest Common Factor

Factoring Out Variables and Powers