Advanced

Tutor model

i62 💰:9.4

i52 💰:7.9

🏆 i67 💰:62.5

i70 💰:125.0

i68 💰:125.0

🏆 i66 💰:75.0

i65 💰:125.0

i63 💰:62.5

i61 💰:75.0

i60 💰:9.4

i60 💰:37.5

i58 💰:62.5

i57 💰:22.5

i57 💰:37.5

i56 💰:45.0

⚡ i55 💰:11.25

i55 💰:37.5

i54 💰:62.5

i52 💰:35.0

i51 💰:37.5

i49 💰:15.0

i49 💰:62.5

i48 💰:11.3

i47 💰:7.5

i44 💰:25.0

i43 💰:37.5

i42 💰:187.5

i40 💰:5.0

i39 💰:100.0

i38 💰:37.5

i37 💰:11.0

i36 💰:187.5

i34 💰:12.5

i33 💰:37.5

i30 💰:1.0

i29 💰:1.5

i28 💰:150.0

i23 💰:20.0

i20 💰:4.0

i18 💰:1.0

i16 💰:25.0

i13 💰:1.5

Keep session private

What would you like to learn?

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

Lesson generation model

System default

i58 💰:18.8

i31 💰:11.0

i25 💰:31.3

Tutoring abilities

When you reduce $\frac{18}{24}$ to $\frac{3}{4}$, you divide both numbers by their common factor of $6$. But what happens when fractions are built out of variables and polynomials instead of plain numbers?

A **rational expression** is a fraction whose numerator and denominator are both polynomials.

This is formally known as the **Fundamental Principle of Rational Expressions**: $\frac{A \cdot C}{B \cdot C} = \frac{A}{B}$, where $B \neq 0$ and $C \neq 0$.

Just as numerical fractions simplify by dividing out common factors (like $\frac{6}{8} = \frac{2 \cdot 3}{2 \cdot 4} = \frac{3}{4}$), rational expressions are simplified using the **Fundamental Principle of Rational Expressions**: $\frac{A \cdot C}{B \cdot C} = \frac{A}{B}$ where $B \neq 0$ and $C \neq 0$.

<!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>Simplifying Rational Expressions</title>
  <link rel="stylesheet" href="{{VENDOR_CDN}}/katex.min.css">
  <script src="{{VENDOR_CDN}}/katex.min.js"></script>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
    }

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

    .toolbar {
      display: flex;
      gap: 8px;
      background: #f1f5f9;
      padding: 4px;
      border-radius: 999px;
    }

    .tab-btn {
      border: none;
      background: transparent;
      padding: 6px 14px;
      font-size: 13px;
      font-weight: 600;
      color: #64748b;
      border-radius: 999px;
      cursor: pointer;
      transition: all 0.2s ease;
    }

    .tab-btn.active {
      background: #ffffff;
      color: #2563eb;
      box-shadow: 0 1px 3px rgba(0, 0, 0, 0.1);
    }

    .stage {
      width: 100%;
      max-width: 500px;
      min-height: 200px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      position: relative;
    }

    .math-card {
      background: #ffffff;
      border: 1.5px solid #e2e8f0;
      border-radius: 16px;
      padding: 24px 20px;
      width: 100%;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 170px;
      box-shadow: 0 4px 6px -1px rgba(0, 0, 0, 0.05);
      position: relative;
    }

    .math-display {
      font-size: 26px;
      min-height: 70px;
      display: flex;
      align-items: center;
      justify-content: center;
      transition: opacity 0.25s ease;
    }

    .badge-container {
      margin-top: 14px;
      height: 28px;
      display: flex;
      align-items: center;
      justify-content: center;
    }

    .step-badge {
      display: inline-flex;
      align-items: center;
      gap: 6px;
      font-size: 13px;
      font-weight: 600;
      padding: 4px 12px;
      border-radius: 999px;
      background: #eff6ff;
      color: #2563eb;
      border: 1px solid #bfdbfe;
    }

    .step-badge.highlight {
      background: #fdf2f8;
      color: #db2777;
      border-color: #fbcfe8;
    }

    .step-badge.success {
      background: #f0fdf4;
      color: #16a34a;
      border-color: #bbf7d0;
    }

    .controls {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 12px;
      width: 100%;
    }

    .stepper-dots {
      display: flex;
      gap: 6px;
      background: #f1f5f9;
      padding: 4px 8px;
      border-radius: 999px;
    }

    .step-btn {
      border: none;
      background: transparent;
      padding: 6px 12px;
      font-size: 12px;
      font-weight: 600;
      color: #64748b;
      border-radius: 999px;
      cursor: pointer;
      transition: all 0.2s ease;
    }

    .step-btn.active {
      background: #2563eb;
      color: #ffffff;
    }

    .play-btn {
      border: 1px solid #e2e8f0;
      background: #ffffff;
      width: 34px;
      height: 34px;
      border-radius: 50%;
      display: flex;
      align-items: center;
      justify-content: center;
      cursor: pointer;
      font-size: 13px;
      color: #334155;
      box-shadow: 0 1px 2px rgba(0, 0, 0, 0.05);
      transition: background 0.2s ease;
    }

    .play-btn:active {
      background: #f8fafc;
    }
  </style>
</head>
<body>

  <div class="toolbar">
    <button class="tab-btn active" id="tabGeneral">Formula</button>
    <button class="tab-btn" id="tabExample">Polynomial</button>
  </div>

  <div class="stage">
    <div class="math-card">
      <div class="math-display" id="mathDisplay"></div>
      <div class="badge-container" id="badgeContainer"></div>
    </div>
  </div>

  <div class="controls">
    <button class="play-btn" id="playBtn" aria-label="Play/Pause">⏸</button>
    <div class="stepper-dots">
      <button class="step-btn active" id="step0">1. Factor</button>
      <button class="step-btn" id="step1">2. Giant 1</button>
      <button class="step-btn" id="step2">3. Simplify</button>
    </div>
  </div>

  <script>
    const data = {
      general: [
        {
          latex: "\\frac{A \\cdot \\mathbf{\\color{#db2777}{C}}}{B \\cdot \\mathbf{\\color{#db2777}{C}}}",
          badge: "Identify Common Factor",
          badgeClass: "highlight"
        },
        {
          latex: "\\frac{A}{B} \\cdot \\left(\\mathbf{\\color{#db2777}{\\frac{C}{C}}}\\right) = \\frac{A}{B} \\cdot \\mathbf{\\color{#2563eb}{1}}",
          badge: "C / C = 1",
          badgeClass: ""
        },
        {
          latex: "\\mathbf{\\frac{A}{B}} \\quad (B, C \\neq 0)",
          badge: "Simplified Form",
          badgeClass: "success"
        }
      ],
      example: [
        {
          latex: "\\frac{(x - 1)\\mathbf{\\color{#db2777}{(x + 4)}}}{(x + 3)\\mathbf{\\color{#db2777}{(x + 4)}}}",
          badge: "Common Factor: (x + 4)",
          badgeClass: "highlight"
        },
        {
          latex: "\\frac{x - 1}{x + 3} \\cdot \\left(\\mathbf{\\color{#db2777}{\\frac{x + 4}{x + 4}}}\\right) = \\frac{x - 1}{x + 3} \\cdot \\mathbf{\\color{#2563eb}{1}}",
          badge: "(x + 4) / (x + 4) = 1",
          badgeClass: ""
        },
        {
          latex: "\\mathbf{\\frac{x - 1}{x + 3}} \\quad (x \\neq -3, -4)",
          badge: "Simplified Form",
          badgeClass: "success"
        }
      ]
    };

    let currentTab = 'general';
    let currentStep = 0;
    let isPlaying = true;
    let timer = null;

    const mathDisplay = document.getElementById('mathDisplay');
    const badgeContainer = document.getElementById('badgeContainer');
    const playBtn = document.getElementById('playBtn');
    const tabGeneral = document.getElementById('tabGeneral');
    const tabExample = document.getElementById('tabExample');
    const stepBtns = [
      document.getElementById('step0'),
      document.getElementById('step1'),
      document.getElementById('step2')
    ];

    function renderMath(str, el) {
      if (window.katex) {
        try {
          katex.render(str, el, { displayMode: true, throwOnError: false });
        } catch (e) {
          el.textContent = str;
        }
      } else {
        el.textContent = str;
      }
    }

    function update() {
      const stepData = data[currentTab][currentStep];

      // Update Math
      mathDisplay.style.opacity = '0';
      setTimeout(() => {
        renderMath(stepData.latex, mathDisplay);
        mathDisplay.style.opacity = '1';
      }, 150);

      // Update Badge
      badgeContainer.innerHTML = `<span class="step-badge ${stepData.badgeClass}">${stepData.badge}</span>`;

      // Update Stepper Buttons
      stepBtns.forEach((btn, idx) => {
        btn.classList.toggle('active', idx === currentStep);
      });

      // Update Tabs
      tabGeneral.classList.toggle('active', currentTab === 'general');
      tabExample.classList.toggle('active', currentTab === 'example');
    }

    function setStep(idx) {
      currentStep = idx;
      update();
    }

    function nextStep() {
      currentStep = (currentStep + 1) % 3;
      update();
    }

    function startAutoPlay() {
      stopAutoPlay();
      timer = setInterval(nextStep, 3500);
      isPlaying = true;
      playBtn.textContent = '⏸';
    }

    function stopAutoPlay() {
      if (timer) clearInterval(timer);
      isPlaying = false;
      playBtn.textContent = '▶';
    }

    function togglePlay() {
      if (isPlaying) {
        stopAutoPlay();
      } else {
        nextStep();
        startAutoPlay();
      }
    }

    // Events
    playBtn.addEventListener('click', togglePlay);

    stepBtns.forEach((btn, idx) => {
      btn.addEventListener('click', () => {
        stopAutoPlay();
        setStep(idx);
      });
    });

    tabGeneral.addEventListener('click', () => {
      currentTab = 'general';
      currentStep = 0;
      update();
    });

    tabExample.addEventListener('click', () => {
      currentTab = 'example';
      currentStep = 0;
      update();
    });

    // Init
    update();
    startAutoPlay();
  </script>
</body>
</html>

How do we find these hidden matching pieces when polynomials are expanded?

To simplify any rational expression, you first **factor completely** both the top and bottom expressions into multiplied binomials or monomials. Once in factored form, you cancel any identical **common factors** shared by both parts.

Let's simplify $\frac{x^2 - 4}{x^2 + 3x + 2}$. The numerator is a difference of squares $(x - 2)(x + 2)$, while the denominator factors into $(x + 1)(x + 2)$.

<!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;
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif;
    }

    body {
      background-color: #f8f9fa;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
      padding: 16px;
    }

    .main-container {
      width: 100%;
      max-width: 580px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 16px;
    }

    /* Step Navigation Pills */
    .step-nav {
      display: flex;
      flex-wrap: wrap;
      justify-content: center;
      gap: 8px;
      width: 100%;
    }

    .step-btn {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 20px;
      padding: 6px 14px;
      font-size: 13px;
      font-weight: 600;
      color: #64748b;
      cursor: pointer;
      display: flex;
      align-items: center;
      gap: 6px;
      transition: all 0.25s ease;
      box-shadow: 0 1px 2px rgba(0,0,0,0.03);
    }

    .step-btn.active {
      background: #2563eb;
      border-color: #2563eb;
      color: #ffffff;
      box-shadow: 0 3px 8px rgba(37, 99, 235, 0.25);
    }

    .stage-card {
      background: #ffffff;
      border-radius: 16px;
      border: 1px solid #e2e8f0;
      width: 100%;
      height: 230px;
      position: relative;
      overflow: hidden;
      box-shadow: 0 4px 12px rgba(0, 0, 0, 0.04);
      display: flex;
      align-items: center;
      justify-content: center;
    }

    .step-panel {
      position: absolute;
      top: 0;
      left: 0;
      width: 100%;
      height: 100%;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      opacity: 0;
      pointer-events: none;
      transition: opacity 0.35s ease, transform 0.35s ease;
      transform: translateY(6px);
      padding: 16px;
    }

    .step-panel.active {
      opacity: 1;
      pointer-events: auto;
      transform: translateY(0);
    }

    .math-display {
      font-size: 28px;
      display: flex;
      align-items: center;
      justify-content: center;
      margin-bottom: 8px;
    }

    .badge-label {
      font-size: 13px;
      font-weight: 600;
      color: #64748b;
      text-transform: uppercase;
      letter-spacing: 0.05em;
      background: #f1f5f9;
      padding: 4px 12px;
      border-radius: 12px;
    }

    /* Highlight common factor */
    .highlight-factor {
      color: #2563eb;
      background: #eff6ff;
      border-radius: 6px;
      padding: 0 4px;
      position: relative;
      display: inline-block;
    }

    .strike-through {
      position: relative;
      color: #94a3b8;
    }

    .strike-through::after {
      content: '';
      position: absolute;
      left: -4px;
      top: 50%;
      width: calc(100% + 8px);
      height: 2.5px;
      background: #ef4444;
      transform: rotate(-15deg);
      border-radius: 2px;
    }

    /* Controls Bar */
    .controls {
      display: flex;
      align-items: center;
      justify-content: space-between;
      width: 100%;
      padding: 0 4px;
    }

    .ctrl-btn {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      color: #1e2945;
      font-size: 14px;
      font-weight: 600;
      border-radius: 8px;
      padding: 8px 16px;
      cursor: pointer;
      display: flex;
      align-items: center;
      gap: 6px;
      transition: background 0.2s, border-color 0.2s;
    }

    .ctrl-btn:hover {
      background: #f8fafc;
      border-color: #cbd5e1;
    }

    .ctrl-btn:disabled {
      opacity: 0.4;
      cursor: not-allowed;
    }

    .dots-indicator {
      display: flex;
      gap: 6px;
    }

    .dot {
      width: 8px;
      height: 8px;
      border-radius: 50%;
      background: #cbd5e1;
      cursor: pointer;
      transition: all 0.25s ease;
    }

    .dot.active {
      background: #2563eb;
      width: 22px;
      border-radius: 4px;
    }

    .katex-display {
      margin: 0 !important;
    }
  </style>
</head>
<body>

<div class="main-container">
  
  <div class="step-nav" id="stepNav">
    <button class="step-btn active" data-step="0">1. Original</button>
    <button class="step-btn" data-step="1">2. Factor</button>
    <button class="step-btn" data-step="2">3. Cancel</button>
    <button class="step-btn" data-step="3">4. Simplified</button>
  </div>

  
  <div class="stage-card">
    
    <div class="step-panel active" id="panel-0">
      <div class="badge-label" style="margin-bottom: 12px;">Given Expression</div>
      <div class="math-display" id="math-step-1"></div>
    </div>

    
    <div class="step-panel" id="panel-1">
      <div class="badge-label" style="margin-bottom: 12px; color: #2563eb; background: #eff6ff;">Factored Form</div>
      <div class="math-display" id="math-step-2"></div>
    </div>

    
    <div class="step-panel" id="panel-2">
      <div class="badge-label" style="margin-bottom: 12px; color: #dc2626; background: #fef2f2;">Divide Common Factor</div>
      <div class="math-display" id="math-step-3"></div>
    </div>

    
    <div class="step-panel" id="panel-3">
      <div class="badge-label" style="margin-bottom: 12px; color: #059669; background: #ecfdf5;">Simplified Result</div>
      <div class="math-display" id="math-step-4"></div>
    </div>
  </div>

  
  <div class="controls">
    <button class="ctrl-btn" id="prevBtn" disabled>← Back</button>
    <div class="dots-indicator" id="dotsContainer">
      <div class="dot active" data-step="0"></div>
      <div class="dot" data-step="1"></div>
      <div class="dot" data-step="2"></div>
      <div class="dot" data-step="3"></div>
    </div>
    <button class="ctrl-btn" id="nextBtn">Next →</button>
  </div>
</div>

<script>
  let currentStep = 0;
  const totalSteps = 4;
  let autoPlayTimer = null;

  // Render KaTeX expressions
  function renderAllMath() {
    try {
      katex.render("\\frac{x^2 - 4}{x^2 + 3x + 2}", document.getElementById('math-step-1'), { displayMode: true });
      katex.render("\\frac{(x - 2)(x + 2)}{(x + 1)(x + 2)}", document.getElementById('math-step-2'), { displayMode: true });
      
      // Step 3 with visual cancellation styling
      const cancelHtml = `
        <span class="katex-display">
          <span class="katex">
            <span class="katex-html">
              <span class="base">
                <span class="mord">
                  <span class="mopen nulldelimiter"></span>
                  <span class="mfrac">
                    <span class="vlist-t vlist-t2">
                      <span class="vlist-r">
                        <span class="vlist" style="height: 1.4em;">
                          <span style="top: -3.34em;">
                            <span class="pstrut" style="height: 3em;"></span>
                            <span class="mord">
                              <span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right: 0.22em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.22em;"></span><span class="mord">2</span><span class="mclose">)</span>
                              <span class="strike-through" style="margin-left: 2px;"><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right: 0.22em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.22em;"></span><span class="mord">2</span><span class="mclose">)</span></span>
                            </span>
                          </span>
                          <span style="top: -2.3em;">
                            <span class="pstrut" style="height: 3em;"></span>
                            <span class="frac-line" style="border-bottom-width: 0.04em;"></span>
                          </span>
                          <span style="top: -1.25em;">
                            <span class="pstrut" style="height: 3em;"></span>
                            <span class="mord">
                              <span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right: 0.22em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.22em;"></span><span class="mord">1</span><span class="mclose">)</span>
                              <span class="strike-through" style="margin-left: 2px;"><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right: 0.22em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.22em;"></span><span class="mord">2</span><span class="mclose">)</span></span>
                            </span>
                          </span>
                        </span>
                        <span class="vlist-s">​</span>
                      </span>
                      <span class="vlist-r"><span class="vlist" style="height: 1.45em;"><span></span></span></span>
                    </span>
                  </span>
                  <span class="mclose nulldelimiter"></span>
                </span>
              </span>
            </span>
          </span>
        </span>
      `;
      document.getElementById('math-step-3').innerHTML = cancelHtml;

      katex.render("\\frac{x - 2}{x + 1}", document.getElementById('math-step-4'), { displayMode: true });
    } catch (e) {
      // Fallback text math if KaTeX is unavailable
      document.getElementById('math-step-1').textContent = "(x² - 4) / (x² + 3x + 2)";
      document.getElementById('math-step-2').textContent = "(x - 2)(x + 2) / (x + 1)(x + 2)";
      document.getElementById('math-step-3').textContent = "(x - 2)[x+2] / (x + 1)[x+2]";
      document.getElementById('math-step-4').textContent = "(x - 2) / (x + 1)";
    }
  }

  function goToStep(idx) {
    currentStep = idx;

    // Update Panels
    document.querySelectorAll('.step-panel').forEach((panel, i) => {
      panel.classList.toggle('active', i === currentStep);
    });

    // Update Nav Buttons
    document.querySelectorAll('.step-btn').forEach((btn, i) => {
      btn.classList.toggle('active', i === currentStep);
    });

    // Update Dots
    document.querySelectorAll('.dot').forEach((dot, i) => {
      dot.classList.toggle('active', i === currentStep);
    });

    // Update Buttons
    document.getElementById('prevBtn').disabled = (currentStep === 0);
    const nextBtn = document.getElementById('nextBtn');
    if (currentStep === totalSteps - 1) {
      nextBtn.textContent = '↺ Restart';
    } else {
      nextBtn.textContent = 'Next →';
    }
  }

  function nextStep() {
    if (currentStep < totalSteps - 1) {
      goToStep(currentStep + 1);
    } else {
      goToStep(0);
    }
  }

  function prevStep() {
    if (currentStep > 0) {
      goToStep(currentStep - 1);
    }
  }

  // Setup Event Handlers
  document.getElementById('nextBtn').addEventListener('click', () => {
    resetAutoPlay();
    nextStep();
  });

  document.getElementById('prevBtn').addEventListener('click', () => {
    resetAutoPlay();
    prevStep();
  });

  document.querySelectorAll('.step-btn').forEach(btn => {
    btn.addEventListener('click', (e) => {
      resetAutoPlay();
      const step = parseInt(e.currentTarget.getAttribute('data-step'), 10);
      goToStep(step);
    });
  });

  document.querySelectorAll('.dot').forEach(dot => {
    dot.addEventListener('click', (e) => {
      resetAutoPlay();
      const step = parseInt(e.currentTarget.getAttribute('data-step'), 10);
      goToStep(step);
    });
  });

  function startAutoPlay() {
    autoPlayTimer = setInterval(() => {
      nextStep();
    }, 4200);
  }

  function resetAutoPlay() {
    if (autoPlayTimer) {
      clearInterval(autoPlayTimer);
      autoPlayTimer = null;
    }
  }

  // Initialize
  renderAllMath();
  startAutoPlay();
</script>

</body>
</html>

Dividing out the matching factor of $(x + 2)$ leaves our simplified answer: $\frac{x - 2}{x + 1}$.

A classic trap is trying to cross out individual added terms before factoring, like slashing the $x^2$ terms in $\frac{x^2 - 4}{x^2 + 2}$. You can only cancel terms that are **multiplied**, never terms that are glued together by addition or subtraction.

What happens when two binomials look almost identical, but their subtraction order is reversed?

When you encounter expressions like $(a - b)$ and $(b - a)$, notice that $(b - a) = -1(a - b)$. Factoring out a **negative one** flips the subtraction order so you can cancel them.

Consider the expression $\frac{2x - 6}{9 - x^2}$. First factor the greatest common factor $2$ from the top to get $2(x - 3)$, and factor the difference of squares on the bottom to get $(3 - x)(3 + x)$.

<!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="{{VENDOR_CDN}}/katex.min.css">
  <script src="{{VENDOR_CDN}}/katex.min.js"></script>
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif;
      background: transparent;
      color: #1e293b;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100vh;
      padding: 16px;
    }

    .diagram {
      width: 100%;
      max-width: 540px;
      display: flex;
      flex-direction: column;
      gap: 16px;
    }

    /* Stepper Navigation */
    .nav-tabs {
      display: flex;
      justify-content: space-between;
      gap: 8px;
      width: 100%;
    }

    .tab-btn {
      flex: 1;
      padding: 8px 4px;
      border: 1px solid #e2e8f0;
      background: #ffffff;
      border-radius: 8px;
      font-size: 13px;
      font-weight: 600;
      color: #64748b;
      cursor: pointer;
      text-align: center;
      transition: all 0.2s ease;
      user-select: none;
    }

    .tab-btn.active {
      background: #eff6ff;
      border-color: #2563eb;
      color: #2563eb;
      box-shadow: 0 1px 3px rgba(37, 99, 235, 0.15);
    }

    /* Main Display Card */
    .stage-card {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 12px;
      padding: 24px 16px;
      min-height: 220px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      position: relative;
      box-shadow: 0 4px 6px -1px rgba(0, 0, 0, 0.05);
    }

    .step-view {
      position: absolute;
      top: 0;
      left: 0;
      right: 0;
      bottom: 0;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      opacity: 0;
      pointer-events: none;
      transition: opacity 0.3s ease, transform 0.3s ease;
      transform: translateY(6px);
      padding: 20px;
    }

    .step-view.active {
      opacity: 1;
      pointer-events: auto;
      transform: translateY(0);
    }

    .badge {
      font-size: 13px;
      font-weight: 600;
      padding: 4px 12px;
      border-radius: 9999px;
      margin-bottom: 18px;
      background: #f1f5f9;
      color: #475569;
    }

    .badge.highlight {
      background: #eff6ff;
      color: #2563eb;
    }

    .math-display {
      font-size: 26px;
      display: flex;
      align-items: center;
      justify-content: center;
    }

    /* Strikethrough style for cancelled factors */
    .cancel-top, .cancel-bot {
      position: relative;
      display: inline-block;
    }

    .cancel-top::after, .cancel-bot::after {
      content: '';
      position: absolute;
      left: -2px;
      right: -2px;
      top: 50%;
      height: 2.5px;
      background: #ef4444;
      transform: rotate(-12deg);
      border-radius: 2px;
    }

    /* Footer Controls */
    .controls {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 12px;
    }

    .btn-ctrl {
      background: #ffffff;
      border: 1px solid #cbd5e1;
      border-radius: 8px;
      padding: 8px 16px;
      font-size: 14px;
      font-weight: 600;
      color: #334155;
      cursor: pointer;
      display: inline-flex;
      align-items: center;
      justify-content: center;
      gap: 6px;
      transition: background 0.15s ease;
      user-select: none;
    }

    .btn-ctrl:active {
      background: #f1f5f9;
    }

    .btn-play {
      min-width: 90px;
      border-color: #2563eb;
      background: #2563eb;
      color: #ffffff;
    }

    .btn-play:active {
      background: #1d4ed8;
    }
  </style>
</head>
<body>

  <div class="diagram">
    
    <div class="nav-tabs" role="tablist">
      <button class="tab-btn active" id="tab0">1. Original</button>
      <button class="tab-btn" id="tab1">2. Factor</button>
      <button class="tab-btn" id="tab2">3. Extract -1</button>
      <button class="tab-btn" id="tab3">4. Simplify</button>
    </div>

    
    <div class="stage-card">
      
      
      <div class="step-view active" id="view0">
        <span class="badge">Original Expression</span>
        <div class="math-display" id="math0"></div>
      </div>

      
      <div class="step-view" id="view1">
        <span class="badge highlight">Factor Terms</span>
        <div class="math-display" id="math1"></div>
      </div>

      
      <div class="step-view" id="view2">
        <span class="badge highlight">Extract -1: (3 - x) = -(x - 3)</span>
        <div class="math-display" id="math2"></div>
      </div>

      
      <div class="step-view" id="view3">
        <span class="badge highlight">Cancel Common Factor</span>
        <div class="math-display" id="math3"></div>
      </div>

    </div>

    
    <div class="controls">
      <button class="btn-ctrl" id="prevBtn">← Prev</button>
      <button class="btn-ctrl btn-play" id="playBtn">Pause ⏸</button>
      <button class="btn-ctrl" id="nextBtn">Next →</button>
    </div>
  </div>

  <script>
    // Safe DOM helper
    const $ = id => document.getElementById(id);

    // Render KaTeX Math cleanly
    katex.render("\\frac{2x - 6}{9 - x^2}", $('math0'), { displayMode: true });
    katex.render("\\frac{2(x - 3)}{(3 - x)(3 + x)}", $('math1'), { displayMode: true });
    katex.render("\\frac{2(x - 3)}{-1(x - 3)(x + 3)}", $('math2'), { displayMode: true });
    katex.render("\\frac{2\\htmlClass{cancel-top}{\\mathbf{(x - 3)}}}{-1\\htmlClass{cancel-bot}{\\mathbf{(x - 3)}}(x + 3)} \\;=\\; -\\frac{2}{x + 3}", $('math3'), {
      displayMode: true,
      trust: true
    });

    const totalSteps = 4;
    let currentStep = 0;
    let autoPlayTimer = null;
    let isPlaying = true;
    const STEP_INTERVAL = 3800;

    function setStep(idx) {
      currentStep = (idx + totalSteps) % totalSteps;
      
      for (let i = 0; i < totalSteps; i++) {
        $('tab' + i).classList.toggle('active', i === currentStep);
        $('view' + i).classList.toggle('active', i === currentStep);
      }
    }

    function nextStep() {
      setStep(currentStep + 1);
    }

    function prevStep() {
      setStep(currentStep - 1);
    }

    function startAutoPlay() {
      stopAutoPlay();
      isPlaying = true;
      $('playBtn').innerHTML = 'Pause ⏸';
      autoPlayTimer = setInterval(nextStep, STEP_INTERVAL);
    }

    function stopAutoPlay() {
      isPlaying = false;
      $('playBtn').innerHTML = 'Play ▶';
      if (autoPlayTimer) {
        clearInterval(autoPlayTimer);
        autoPlayTimer = null;
      }
    }

    function toggleAutoPlay() {
      if (isPlaying) {
        stopAutoPlay();
      } else {
        nextStep();
        startAutoPlay();
      }
    }

    // Attach listeners
    for (let i = 0; i < totalSteps; i++) {
      $('tab' + i).addEventListener('click', () => {
        setStep(i);
        stopAutoPlay();
      });
    }

    $('prevBtn').addEventListener('click', () => {
      prevStep();
      stopAutoPlay();
    });

    $('nextBtn').addEventListener('click', () => {
      nextStep();
      stopAutoPlay();
    });

    $('playBtn').addEventListener('click', toggleAutoPlay);

    // Initial autoplay start
    startAutoPlay();
  </script>
</body>
</html>

Canceling the common factor $(x - 3)$ leaves $\frac{2}{-1(3 + x)}$, which simplifies cleanly to $-\frac{2}{x + 3}$.

Before you ever strike out a term, ask yourself: 'Is this whole piece being multiplied by everything else?' If the answer is yes, divide it out with confidence.

math

Simplifying Rational Expressions. Simplify a rational expression by factoring the numerator and denominator and canceling common factors.

Simplifying Rational Expressions Introduction

Simplify Rational Expressions (Algebra 1 / Algebra 2 Skill Practice)

Simplifying Rational Expressions - Factoring and Canceling Explained

To achieve high conceptual retention in grades 9-10, always begin instruction by connecting polynomial simplification to elementary fraction reduction. Emphasize the core diagnostic rule: 'Factor First, Cancel Second.' Address the persistent error of canceling additive terms by having students physically box or highlight entire factors before drawing cancellation lines. Explicitly mandate recording domain restrictions before crossing out any common factors.

Simplifying Rational Expressions

The Factoring Method