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Override for the tutor's lesson content generation calls. Auto uses the system default (Claude Sonnet 5 for lesson text, Opus 5 for diagrams).

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System default

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Tutoring abilities

If you multiply $3x(2x + 5)$, the distributive property gives you $6x^2 + 15x$.

Factoring is simply hitting **rewind** on multiplication to break an expanded expression back into a product of simpler pieces.

<!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">
<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>
  * { box-sizing: border-box; margin: 0; padding: 0; }
  body {
    font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
    color: #1e2945;
    background: transparent;
    display: flex;
    flex-direction: column;
    align-items: center;
    justify-content: center;
    min-height: 100%;
    padding: 12px;
    user-select: none;
    -webkit-user-select: none;
  }

  .main-wrap {
    width: 100%;
    max-width: 480px;
    display: flex;
    flex-direction: column;
    align-items: center;
    gap: 16px;
  }

  /* Nav Tabs */
  .nav-tabs {
    display: flex;
    gap: 8px;
    background: #f1f5f9;
    padding: 4px;
    border-radius: 10px;
    width: 100%;
    max-width: 380px;
  }
  .nav-btn {
    flex: 1;
    border: none;
    background: transparent;
    padding: 8px 6px;
    font-size: 13px;
    font-weight: 600;
    color: #64748b;
    border-radius: 8px;
    cursor: pointer;
    transition: all 0.25s ease;
    text-align: center;
  }
  .nav-btn.active {
    background: #ffffff;
    color: #4f46e5;
    box-shadow: 0 1px 3px rgba(0,0,0,0.1);
  }

  /* Visual Container */
  .stage-card {
    width: 100%;
    background: #ffffff;
    border: 1px solid #e2e8f0;
    border-radius: 14px;
    padding: 18px 14px;
    display: flex;
    flex-direction: column;
    align-items: center;
    gap: 16px;
    box-shadow: 0 2px 8px rgba(0,0,0,0.04);
  }

  /* Geometric Area Model */
  .area-model {
    display: flex;
    align-items: center;
    justify-content: center;
    gap: 8px;
    width: 100%;
  }

  .side-label {
    font-size: 16px;
    font-weight: 700;
    color: #4f46e5;
    padding: 6px 10px;
    background: #eef2ff;
    border: 2px dashed #4f46e5;
    border-radius: 8px;
    transition: all 0.35s ease;
    opacity: 0.2;
    transform: scale(0.9);
  }
  .side-label.visible {
    opacity: 1;
    transform: scale(1);
    border-style: solid;
  }

  .box-group {
    display: flex;
    flex-direction: column;
    gap: 6px;
  }

  .top-labels {
    display: flex;
    gap: 6px;
    height: 24px;
    align-items: center;
  }
  .top-label {
    font-size: 14px;
    font-weight: 600;
    color: #0284c7;
    text-align: center;
    opacity: 0;
    transition: opacity 0.35s ease;
  }
  .top-label.w1 { width: 110px; }
  .top-label.w2 { width: 80px; }
  .top-label.visible { opacity: 1; }

  .rectangles {
    display: flex;
    gap: 6px;
  }

  .rect {
    height: 74px;
    border-radius: 8px;
    display: flex;
    flex-direction: column;
    align-items: center;
    justify-content: center;
    transition: all 0.35s ease;
    border: 2px solid transparent;
  }

  .rect-1 {
    width: 110px;
    background: #eff6ff;
    border-color: #93c5fd;
    color: #1e40af;
  }

  .rect-2 {
    width: 80px;
    background: #f0fdf4;
    border-color: #86efac;
    color: #166534;
  }

  .rect.highlight {
    box-shadow: 0 0 0 3px rgba(79, 70, 229, 0.25);
    border-color: #4f46e5;
  }

  /* Math Equations Area */
  .equation-box {
    display: flex;
    align-items: center;
    justify-content: center;
    min-height: 48px;
    width: 100%;
    background: #f8fafc;
    border-radius: 10px;
    padding: 8px 12px;
    font-size: 19px;
  }

  /* Direction Indicator */
  .direction-banner {
    display: flex;
    align-items: center;
    justify-content: center;
    gap: 8px;
    font-size: 13px;
    font-weight: 700;
    color: #4f46e5;
    letter-spacing: 0.5px;
    text-transform: uppercase;
  }

  .tag-gcf {
    color: #4f46e5;
    background: #e0e7ff;
    padding: 2px 6px;
    border-radius: 5px;
    font-weight: 700;
  }
</style>
</head>
<body>

<div class="main-wrap">
  <div class="nav-tabs">
    <button class="nav-btn active" id="btn-1" onclick="setStep(1)">1. Expanded</button>
    <button class="nav-btn" id="btn-2" onclick="setStep(2)">2. Find GCF</button>
    <button class="nav-btn" id="btn-3" onclick="setStep(3)">3. Rewind (Factor)</button>
  </div>

  <div class="stage-card">
    <div class="direction-banner" id="direction-text">
      <span>Expanded Terms</span>
    </div>

    
    <div class="area-model">
      <div class="side-label" id="side-gcf">2x</div>
      
      <div class="box-group">
        <div class="top-labels">
          <div class="top-label w1" id="top-l1">3x</div>
          <div class="top-label w2" id="top-l2">+ 4</div>
        </div>
        <div class="rectangles">
          <div class="rect rect-1" id="box1">
            <span id="box1-math"></span>
          </div>
          <div class="rect rect-2" id="box2">
            <span id="box2-math"></span>
          </div>
        </div>
      </div>
    </div>

    
    <div class="equation-box">
      <span id="main-eq"></span>
    </div>
  </div>
</div>

<script>
  let currentStep = 1;
  let autoTimer = null;

  function renderMath() {
    const box1 = document.getElementById('box1-math');
    const box2 = document.getElementById('box2-math');
    const mainEq = document.getElementById('main-eq');
    const sideGcf = document.getElementById('side-gcf');
    const topL1 = document.getElementById('top-l1');
    const topL2 = document.getElementById('top-l2');

    katex.render("2x", sideGcf);
    katex.render("3x", topL1);
    katex.render("+ 4", topL2);

    if (currentStep === 1) {
      katex.render("6x^2", box1);
      katex.render("8x", box2);
      katex.render("6x^2 + 8x", mainEq);
    } else if (currentStep === 2) {
      katex.render("\\mathbf{2x} \\cdot 3x", box1);
      katex.render("\\mathbf{2x} \\cdot 4", box2);
      katex.render("(\\mathbf{2x} \\cdot 3x) + (\\mathbf{2x} \\cdot 4)", mainEq);
    } else {
      katex.render("3x", box1);
      katex.render("4", box2);
      katex.render("\\mathbf{2x}(3x + 4)", mainEq);
    }
  }

  function setStep(step) {
    currentStep = step;
    clearInterval(autoTimer);

    // Update Nav
    for (let i = 1; i <= 3; i++) {
      document.getElementById(`btn-${i}`).classList.toggle('active', i === step);
    }

    const sideGcf = document.getElementById('side-gcf');
    const topL1 = document.getElementById('top-l1');
    const topL2 = document.getElementById('top-l2');
    const box1 = document.getElementById('box1');
    const box2 = document.getElementById('box2');
    const dirText = document.getElementById('direction-text');

    if (step === 1) {
      sideGcf.classList.remove('visible');
      topL1.classList.remove('visible');
      topL2.classList.remove('visible');
      box1.classList.remove('highlight');
      box2.classList.remove('highlight');
      dirText.innerHTML = "<span>Expanded Form</span>";
    } else if (step === 2) {
      sideGcf.classList.add('visible');
      topL1.classList.remove('visible');
      topL2.classList.remove('visible');
      box1.classList.add('highlight');
      box2.classList.add('highlight');
      dirText.innerHTML = "<span>Common Factor: <span class='tag-gcf'>2x</span></span>";
    } else {
      sideGcf.classList.add('visible');
      topL1.classList.add('visible');
      topL2.classList.add('visible');
      box1.classList.remove('highlight');
      box2.classList.remove('highlight');
      dirText.innerHTML = "<span>⏪ Factored (Rewound)</span>";
    }

    renderMath();
  }

  window.addEventListener('DOMContentLoaded', () => {
    renderMath();
    setStep(1);

    // Pedagogical auto-cycle (4s per step)
    autoTimer = setInterval(() => {
      let next = currentStep + 1;
      if (next > 3) next = 1;
      setStep(next);
    }, 4000);
  });
</script>

</body>
</html>

How do we know what piece to pull out of every term?

The **greatest common factor (GCF)** is the largest expression that divides evenly into every single term of a polynomial.

To find the GCF, you combine two parts: the greatest common numerical factor of the **coefficients** (the numbers multiplying the variables) and the smallest exponent of each shared variable.

<!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">
<style>
  * { box-sizing: border-box; margin: 0; padding: 0; }
  body {
    font-family: system-ui, -apple-system, sans-serif;
    background: transparent;
    color: #1e2945;
    display: flex;
    flex-direction: column;
    align-items: center;
    justify-content: center;
    min-height: 100vh;
    padding: 12px;
  }

  .diagram {
    width: 100%;
    max-width: 480px;
    display: flex;
    flex-direction: column;
    gap: 12px;
  }

  /* Nav Tabs */
  .tabs {
    display: flex;
    justify-content: center;
    gap: 8px;
  }
  .tab-btn {
    border: 1px solid #e2e8f0;
    background: #ffffff;
    padding: 6px 14px;
    border-radius: 20px;
    font-size: 13px;
    font-weight: 600;
    color: #64748b;
    cursor: pointer;
    transition: all 0.2s ease;
  }
  .tab-btn.active {
    background: #2563eb;
    border-color: #2563eb;
    color: #ffffff;
  }

  /* Expression Stage */
  .stage {
    background: #ffffff;
    border: 1px solid #e2e8f0;
    border-radius: 12px;
    padding: 16px;
    display: flex;
    flex-direction: column;
    align-items: center;
    gap: 16px;
  }

  .expr-box {
    font-size: 26px;
    font-weight: 700;
    letter-spacing: 1px;
    display: flex;
    align-items: center;
    justify-content: center;
    gap: 6px;
  }

  .num {
    padding: 2px 6px;
    border-radius: 6px;
    transition: all 0.3s ease;
  }
  .var {
    padding: 2px 6px;
    border-radius: 6px;
    transition: all 0.3s ease;
  }
  sup { font-size: 0.65em; vertical-align: super; }

  .hl-num { background: #fef3c7; color: #d97706; }
  .hl-var { background: #dbeafe; color: #2563eb; }

  /* Breakdown Grid */
  .grid-breakdown {
    display: grid;
    grid-template-columns: 1fr 1fr;
    gap: 10px;
    width: 100%;
  }

  .sub-card {
    border: 1px solid #e2e8f0;
    border-radius: 10px;
    padding: 10px;
    display: flex;
    flex-direction: column;
    align-items: center;
    gap: 6px;
    background: #f8fafc;
    transition: all 0.3s ease;
  }
  .sub-card.active-num {
    border-color: #d97706;
    background: #fffbeb;
  }
  .sub-card.active-var {
    border-color: #2563eb;
    background: #eff6ff;
  }

  .label-tag {
    font-size: 11px;
    font-weight: 700;
    text-transform: uppercase;
    letter-spacing: 0.5px;
    color: #64748b;
  }
  .sub-card.active-num .label-tag { color: #d97706; }
  .sub-card.active-var .label-tag { color: #2563eb; }

  .sub-calc {
    font-size: 15px;
    font-weight: 600;
    color: #475569;
  }
  .sub-result {
    font-size: 18px;
    font-weight: 700;
  }
  .res-num { color: #d97706; }
  .res-var { color: #2563eb; }

  /* Result Box */
  .result-banner {
    width: 100%;
    padding: 12px;
    border-radius: 10px;
    background: #f1f5f9;
    border: 1px solid #e2e8f0;
    display: flex;
    justify-content: space-around;
    align-items: center;
    transition: all 0.3s ease;
  }
  .res-item {
    display: flex;
    flex-direction: column;
    align-items: center;
    gap: 2px;
  }
  .res-val {
    font-size: 20px;
    font-weight: 800;
  }
  .res-gcf {
    color: #0f172a;
    display: flex;
    align-items: center;
  }

  /* Controls */
  .stepper {
    display: flex;
    justify-content: space-between;
    align-items: center;
    gap: 8px;
  }
  .step-dots {
    display: flex;
    gap: 6px;
  }
  .dot {
    width: 8px;
    height: 8px;
    border-radius: 50%;
    background: #cbd5e1;
    cursor: pointer;
    transition: all 0.2s ease;
  }
  .dot.active {
    background: #2563eb;
    width: 20px;
    border-radius: 4px;
  }
  .btn-ctrl {
    border: 1px solid #e2e8f0;
    background: #ffffff;
    padding: 6px 14px;
    border-radius: 8px;
    font-size: 13px;
    font-weight: 600;
    color: #1e2945;
    cursor: pointer;
  }
</style>
</head>
<body>

<div class="diagram">
  
  <div class="tabs">
    <button class="tab-btn active" onclick="setExample(0)">6x³ + 9x²</button>
    <button class="tab-btn" onclick="setExample(1)">12y⁴ − 8y²</button>
    <button class="tab-btn" onclick="setExample(2)">15a⁵ + 10a³</button>
  </div>

  
  <div class="stage">
    
    <div class="expr-box" id="exprBox"></div>

    
    <div class="grid-breakdown">
      
      <div class="sub-card" id="cardNum">
        <span class="label-tag">Coefficients</span>
        <span class="sub-calc" id="calcNum"></span>
        <span class="sub-result res-num" id="resNum"></span>
      </div>
      
      <div class="sub-card" id="cardVar">
        <span class="label-tag">Smallest Power</span>
        <span class="sub-calc" id="calcVar"></span>
        <span class="sub-result res-var" id="resVar"></span>
      </div>
    </div>

    
    <div class="result-banner" id="resBanner">
      <div class="res-item">
        <span class="label-tag">GCF</span>
        <span class="res-val res-gcf" id="finalGcf"></span>
      </div>
      <div class="res-item">
        <span class="label-tag">Factored</span>
        <span class="res-val" style="color:#059669;" id="finalFactored"></span>
      </div>
    </div>
  </div>

  
  <div class="stepper">
    <button class="btn-ctrl" onclick="prevStep()">←</button>
    <div class="step-dots" id="dots">
      <div class="dot active" onclick="goStep(0)"></div>
      <div class="dot" onclick="goStep(1)"></div>
      <div class="dot" onclick="goStep(2)"></div>
      <div class="dot" onclick="goStep(3)"></div>
    </div>
    <button class="btn-ctrl" onclick="nextStep()">→</button>
  </div>
</div>

<script>
const data = [
  {
    c1: 6, c2: 9, op: "+", v: "x", p1: 3, p2: 2,
    gcfNum: 3, gcfVar: "x²",
    factored: "3x²(2x + 3)"
  },
  {
    c1: 12, c2: 8, op: "−", v: "y", p1: 4, p2: 2,
    gcfNum: 4, gcfVar: "y²",
    factored: "4y²(3y² − 2)"
  },
  {
    c1: 15, c2: 10, op: "+", v: "a", p1: 5, p2: 3,
    gcfNum: 5, gcfVar: "a³",
    factored: "5a³(3a² + 2)"
  }
];

let curEx = 0;
let curStep = 0;

function render() {
  const d = data[curEx];
  
  // Expression with dynamic highlights based on step
  const nClass = (curStep === 1 || curStep >= 3) ? 'num hl-num' : 'num';
  const vClass = (curStep === 2 || curStep >= 3) ? 'var hl-var' : 'var';

  document.getElementById('exprBox').innerHTML = `
    <span class="${nClass}">${d.c1}</span><span class="${vClass}">${d.v}<sup>${d.p1}</sup></span>
    <span>${d.op}</span>
    <span class="${nClass}">${d.c2}</span><span class="${vClass}">${d.v}<sup>${d.p2}</sup></span>
  `;

  // Number card
  const cNum = document.getElementById('cardNum');
  cNum.className = (curStep === 1 || curStep >= 3) ? 'sub-card active-num' : 'sub-card';
  document.getElementById('calcNum').textContent = `${d.c1}, ${d.c2}`;
  document.getElementById('resNum').textContent = (curStep >= 1) ? `→ ${d.gcfNum}` : '?';

  // Variable card
  const cVar = document.getElementById('cardVar');
  cVar.className = (curStep === 2 || curStep >= 3) ? 'sub-card active-var' : 'sub-card';
  document.getElementById('calcVar').textContent = `${d.v}<sup>${d.p1}</sup>, ${d.v}<sup>${d.p2}</sup>`.replace(/<sup>|<\/sup>/g, '');
  document.getElementById('resVar').innerHTML = (curStep >= 2) ? `→ ${d.gcfVar}` : '?';

  // Combined Results
  const showGCF = curStep >= 3;
  document.getElementById('finalGcf').innerHTML = showGCF ? `<span class="res-num">${d.gcfNum}</span><span class="res-var">${d.gcfVar}</span>` : '—';
  document.getElementById('finalFactored').innerHTML = (curStep === 3) ? d.factored : '—';

  // Dots
  const dots = document.getElementById('dots').children;
  for (let i = 0; i < dots.length; i++) {
    dots[i].className = i === curStep ? 'dot active' : 'dot';
  }

  // Tabs
  const tabs = document.querySelectorAll('.tab-btn');
  tabs.forEach((t, i) => {
    t.className = i === curEx ? 'tab-btn active' : 'tab-btn';
  });
}

function setExample(idx) {
  curEx = idx;
  curStep = 0;
  render();
}

function goStep(s) {
  curStep = s;
  render();
}

function nextStep() {
  curStep = (curStep + 1) % 4;
  render();
}

function prevStep() {
  curStep = (curStep - 1 + 4) % 4;
  render();
}

render();
</script>
</body>
</html>

Why do we always choose the smallest exponent? Because $x^2$ can divide into $x^3$ without leaving a fraction, but $x^3$ cannot divide into $x^2$.

Now that we can pinpoint the GCF, how do we factor an entire three-term polynomial?

Let's factor the expression $12x^3 - 8x^2 + 20x$ completely.

<!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>Factoring Out the GCF</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>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
    }

    body {
      padding: 12px;
      color: #1e293b;
      background: transparent;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
    }

    .main-stage {
      width: 100%;
      max-width: 580px;
      display: flex;
      flex-direction: column;
      gap: 12px;
    }

    /* Stepper Controls */
    .controls-row {
      display: flex;
      justify-content: space-between;
      align-items: center;
      gap: 8px;
    }

    .step-tabs {
      display: flex;
      gap: 6px;
      flex: 1;
    }

    .step-tab {
      flex: 1;
      padding: 6px 4px;
      background: #f1f5f9;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
      font-size: 13px;
      font-weight: 600;
      color: #64748b;
      text-align: center;
      cursor: pointer;
      transition: all 0.25s ease;
      user-select: none;
    }

    .step-tab.active {
      background: #059669;
      color: #ffffff;
      border-color: #059669;
      box-shadow: 0 2px 4px rgba(5, 150, 105, 0.2);
    }

    .play-btn {
      padding: 6px 12px;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
      font-size: 14px;
      cursor: pointer;
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To remove the GCF, divide each term of the polynomial by the GCF using the **Quotient Rule of Exponents** ($\frac{x^m}{x^n} = x^{m - n}$).

For our polynomial, this yields $\frac{12x^3}{4x} = 3x^2$, $\frac{-8x^2}{4x} = -2x$, and $\frac{20x}{4x} = 5$.

The fully factored form is $4x(3x^2 - 2x + 5)$.

What happens when a term inside the polynomial looks identical to the GCF itself?

A common mistake is thinking a term disappears when it matches the GCF.

When factoring $6x^2 + 2x$, the GCF is $2x$. Dividing the second term gives $\frac{2x}{2x} = 1$, so the correct answer is $2x(3x + 1)$, never $2x(3x)$.

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Additionally, when the leading term is negative, like $-5x^2 + 15x$, factor out a **negative GCF** ($-5x$) to leave a positive leading term inside: $-5x(x - 3)$.

Can we use this exact same strategy if an expression has multiple different variables?

Consider the polynomial $14a^3b^2 - 21a^2b^4$.

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    <span class="hl-c">14</span><span class="hl-a">a³</span><span class="hl-b">b²</span>
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We find the GCF of the numbers ($7$), the lowest power of $a$ ($a^2$), and the lowest power of $b$ ($b^2$), yielding a total GCF of $7a^2b^2$.

Dividing each term leaves $7a^2b^2(2a - 3b^2)$ — and you can always verify your answer instantly by multiplying the terms back out.

math

Factoring Out the Greatest Common Factor. Factor polynomials by identifying and removing the greatest common factor.

Factoring Polynomials by Taking a Common Factor

Factor Out a Monomial (Algebra 1)

6.2: Greatest Common Factor and Factor by Grouping

Instruction should establish factoring as the explicit inverse of polynomial multiplication via the distributive property. Introduce the lesson with the reverse area model to ground conceptual understanding before moving to algorithmic exponent subtraction rules. Address the $1$-as-a-placeholder misconception early using explicit division steps ($rac{kx^n}{kx^n} = 1$). Always require students to verify results by distributing back to the original polynomial.

Factoring Out the Greatest Common Factor

Finding the Greatest Common Factor

Step-by-Step: Factoring a Trinomial