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Every smooth curve you see in a 3D video game — from the arc of a thrown basketball to the slope of a race track — is built out of algebraic building blocks called **polynomials**.

Before a computer can render those curves, it needs to break the math down into individual parts.

A **term** is a single mathematical unit made of a constant number, one or more variables multiplied together, or both. In any term, the multiplying number in front is the **coefficient**, and the power on the variable is the **exponent**.

<!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>Anatomy of a Polynomial Term</title>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif;
      user-select: none;
      -webkit-user-select: none;
    }

    body {
      background: #f8f9fa;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100%;
      padding: 16px;
      gap: 12px;
    }

    /* Term Selector Tabs */
    .selector-bar {
      display: flex;
      flex-wrap: wrap;
      justify-content: center;
      gap: 8px;
      width: 100%;
      max-width: 520px;
    }

    .tab-btn {
      background: #ffffff;
      border: 2px solid #e2e8f0;
      border-radius: 20px;
      padding: 6px 14px;
      font-size: 15px;
      font-weight: 600;
      color: #64748b;
      cursor: pointer;
      transition: all 0.25s ease;
      outline: none;
    }

    .tab-btn:hover {
      border-color: #cbd5e1;
      color: #1e2945;
    }

    .tab-btn.active {
      background: #1e2945;
      border-color: #1e2945;
      color: #ffffff;
      box-shadow: 0 4px 10px rgba(30, 41, 69, 0.15);
    }

    /* Main Display Stage */
    .stage {
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      width: 100%;
      max-width: 520px;
      background: #ffffff;
      border-radius: 16px;
      padding: 24px 16px 20px;
      box-shadow: 0 2px 8px rgba(0, 0, 0, 0.04);
      position: relative;
    }

    /* Large Equation Representation */
    .math-display {
      display: flex;
      align-items: baseline;
      justify-content: center;
      font-size: 58px;
      font-weight: 700;
      line-height: 1;
      margin-bottom: 24px;
      cursor: pointer;
    }

    .part {
      position: relative;
      display: inline-flex;
      align-items: center;
      justify-content: center;
      padding: 4px 6px;
      border-radius: 8px;
      transition: all 0.2s ease;
    }

    .part-coeff {
      color: #0284c7;
      background: rgba(2, 132, 199, 0.08);
    }

    .part-coeff.highlight {
      background: rgba(2, 132, 199, 0.22);
      box-shadow: 0 0 0 2px #0284c7;
    }

    .part-var {
      color: #059669;
      font-style: italic;
      font-family: "Times New Roman", Times, serif;
      background: rgba(5, 150, 105, 0.08);
      margin-left: 2px;
    }

    .part-var.highlight {
      background: rgba(5, 150, 105, 0.22);
      box-shadow: 0 0 0 2px #059669;
    }

    .part-exp {
      color: #7c3aed;
      font-size: 34px;
      transform: translateY(-20px);
      background: rgba(124, 58, 237, 0.08);
      margin-left: 2px;
    }

    .part-exp.highlight {
      background: rgba(124, 58, 237, 0.22);
      box-shadow: 0 0 0 2px #7c3aed;
    }

    .ghost-note {
      font-size: 13px;
      font-weight: 600;
      color: #94a3b8;
      position: absolute;
      bottom: -18px;
      white-space: nowrap;
    }

    /* Breakdown Component Cards */
    .cards-grid {
      display: grid;
      grid-template-columns: repeat(3, 1fr);
      gap: 10px;
      width: 100%;
    }

    .part-card {
      background: #f8fafc;
      border-radius: 12px;
      padding: 10px 8px;
      text-align: center;
      cursor: pointer;
      transition: all 0.2s ease;
      border: 2px solid transparent;
    }

    .part-card.coeff {
      border-color: rgba(2, 132, 199, 0.2);
    }
    .part-card.coeff:hover, .part-card.coeff.active {
      border-color: #0284c7;
      background: rgba(2, 132, 199, 0.08);
    }

    .part-card.var {
      border-color: rgba(5, 150, 105, 0.2);
    }
    .part-card.var:hover, .part-card.var.active {
      border-color: #059669;
      background: rgba(5, 150, 105, 0.08);
    }

    .part-card.exp {
      border-color: rgba(124, 58, 237, 0.2);
    }
    .part-card.exp:hover, .part-card.exp.active {
      border-color: #7c3aed;
      background: rgba(124, 58, 237, 0.08);
    }

    .card-label {
      font-size: 11px;
      font-weight: 700;
      text-transform: uppercase;
      letter-spacing: 0.5px;
      margin-bottom: 4px;
    }

    .coeff .card-label { color: #0284c7; }
    .var .card-label { color: #059669; }
    .exp .card-label { color: #7c3aed; }

    .card-val {
      font-size: 18px;
      font-weight: 700;
      color: #1e2945;
    }

    /* Summary Bar (Degree) */
    .degree-bar {
      display: flex;
      align-items: center;
      justify-content: space-between;
      width: 100%;
      max-width: 520px;
      background: #ffffff;
      border-radius: 12px;
      padding: 10px 16px;
      box-shadow: 0 1px 4px rgba(0,0,0,0.04);
    }

    .degree-label {
      font-size: 14px;
      font-weight: 600;
      color: #64748b;
    }

    .degree-pill {
      background: #7c3aed;
      color: #ffffff;
      font-size: 13px;
      font-weight: 700;
      padding: 4px 12px;
      border-radius: 20px;
      display: flex;
      align-items: center;
      gap: 4px;
    }

    @media (max-width: 400px) {
      .math-display {
        font-size: 46px;
      }
      .part-exp {
        font-size: 26px;
        transform: translateY(-16px);
      }
      .tab-btn {
        padding: 4px 10px;
        font-size: 13px;
      }
      .card-val {
        font-size: 15px;
      }
    }
  </style>
</head>
<body>

  
  <div class="selector-bar">
    <button class="tab-btn active" onclick="setPreset(0)">7x³</button>
    <button class="tab-btn" onclick="setPreset(1)">−4y⁵</button>
    <button class="tab-btn" onclick="setPreset(2)">x</button>
    <button class="tab-btn" onclick="setPreset(3)">8</button>
    <button class="tab-btn" onclick="setPreset(4)">½a⁴</button>
  </div>

  
  <div class="stage">
    <div class="math-display">
      <div class="part part-coeff" id="disp-coeff" onclick="highlightPart('coeff')">
        7
      </div>
      <div class="part part-var" id="disp-var" onclick="highlightPart('var')">
        x
      </div>
      <div class="part part-exp" id="disp-exp" onclick="highlightPart('exp')">
        3
      </div>
    </div>

    
    <div class="cards-grid">
      <div class="part-card coeff" id="card-coeff" onclick="highlightPart('coeff')">
        <div class="card-label">Coefficient</div>
        <div class="card-val" id="val-coeff">7</div>
      </div>
      <div class="part-card var" id="card-var" onclick="highlightPart('var')">
        <div class="card-label">Variable</div>
        <div class="card-val" id="val-var">x</div>
      </div>
      <div class="part-card exp" id="card-exp" onclick="highlightPart('exp')">
        <div class="card-label">Exponent</div>
        <div class="card-val" id="val-exp">3</div>
      </div>
    </div>
  </div>

  
  <div class="degree-bar">
    <span class="degree-label">Degree of this term:</span>
    <div class="degree-pill" id="val-degree">Degree 3</div>
  </div>

  <script>
    const presets = [
      { coeff: '7', var: 'x', exp: '3', coeffVal: '7', varVal: 'x', expVal: '3', degree: 'Degree 3', hideVar: false, hideExp: false },
      { coeff: '−4', var: 'y', exp: '5', coeffVal: '−4', varVal: 'y', expVal: '5', degree: 'Degree 5', hideVar: false, hideExp: false },
      { coeff: '1', var: 'x', exp: '1', coeffVal: '1 (implied)', varVal: 'x', expVal: '1 (implied)', degree: 'Degree 1', hideCoeffText: '', ghostCoeff: '1', ghostExp: '1' },
      { coeff: '8', var: 'x⁰', exp: '0', coeffVal: '8', varVal: 'none', expVal: '0', degree: 'Degree 0 (Constant)', hideVarDisplay: true },
      { coeff: '½', var: 'a', exp: '4', coeffVal: '½', varVal: 'a', expVal: '4', degree: 'Degree 4', hideVar: false, hideExp: false }
    ];

    let currentPreset = 0;
    let activeHighlight = null;

    function setPreset(index) {
      currentPreset = index;
      const data = presets[index];

      // Update tabs
      document.querySelectorAll('.tab-btn').forEach((btn, i) => {
        btn.classList.toggle('active', i === index);
      });

      // Update Math Display
      const dispCoeff = document.getElementById('disp-coeff');
      const dispVar = document.getElementById('disp-var');
      const dispExp = document.getElementById('disp-exp');

      if (data.hideVarDisplay) {
        dispCoeff.textContent = data.coeff;
        dispVar.style.display = 'none';
        dispExp.style.display = 'none';
      } else if (data.ghostCoeff) {
        dispCoeff.textContent = '1';
        dispCoeff.style.opacity = '0.4';
        dispVar.style.display = 'inline-flex';
        dispVar.textContent = data.var;
        dispExp.style.display = 'inline-flex';
        dispExp.textContent = '1';
        dispExp.style.opacity = '0.4';
      } else {
        dispCoeff.textContent = data.coeff;
        dispCoeff.style.opacity = '1';
        dispVar.style.display = 'inline-flex';
        dispVar.textContent = data.var;
        dispExp.style.display = 'inline-flex';
        dispExp.textContent = data.exp;
        dispExp.style.opacity = '1';
      }

      // Update Breakdown Cards
      document.getElementById('val-coeff').textContent = data.coeffVal;
      document.getElementById('val-var').textContent = data.varVal;
      document.getElementById('val-exp').textContent = data.expVal;

      // Update Degree
      document.getElementById('val-degree').textContent = data.degree;

      // Reset highlights
      clearHighlights();
    }

    function highlightPart(part) {
      if (activeHighlight === part) {
        clearHighlights();
        return;
      }
      activeHighlight = part;

      ['coeff', 'var', 'exp'].forEach(p => {
        const isMatch = (p === part);
        document.getElementById(`disp-${p}`).classList.toggle('highlight', isMatch);
        document.getElementById(`card-${p}`).classList.toggle('active', isMatch);
      });
    }

    function clearHighlights() {
      activeHighlight = null;
      ['coeff', 'var', 'exp'].forEach(p => {
        document.getElementById(`disp-${p}`).classList.remove('highlight');
        document.getElementById(`card-${p}`).classList.remove('active');
      });
    }
  </script>
</body>
</html>

What happens when we string several of these terms together using addition and subtraction?

A **polynomial** is a finite sum or difference of terms of the form $a x^n$, where $a$ is a real number and the exponent $n$ is a non-negative integer ($n \in \{0, 1, 2, 3, \dots\}$).

Expressions with negative exponents (such as $x^{-1} = \frac{1}{x}$) or rational non-integer exponents (such as $x^{1/2} = \sqrt{x}$) are **not polynomials**.

Expressions containing negative exponents (such as $x^{-1} = \frac{1}{x}$) or rational non-integer exponents (such as $x^{1/2} = \sqrt{x}$) are **not** polynomials.

<!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>Polynomial Identifier</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; }
    body {
      font-family: system-ui, -apple-system, sans-serif;
      color: #1e2945;
      background: transparent;
      padding: 8px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 12px;
      user-select: none;
    }
    .tabs {
      display: flex;
      flex-wrap: wrap;
      gap: 6px;
      justify-content: center;
      width: 100%;
      max-width: 540px;
    }
    .tab-btn {
      background: #ffffff;
      border: 1px solid #d1d5db;
      border-radius: 8px;
      padding: 6px 12px;
      font-size: 14px;
      cursor: pointer;
      color: #4b5563;
      transition: all 0.2s ease;
      display: flex;
      align-items: center;
      gap: 4px;
    }
    .tab-btn:hover {
      background: #f3f4f6;
    }
    .tab-btn.active {
      background: #22b7ff;
      border-color: #22b7ff;
      color: #ffffff;
      font-weight: 600;
    }
    .display-panel {
      width: 100%;
      max-width: 540px;
      background: #ffffff;
      border: 1px solid #e5e7eb;
      border-radius: 12px;
      padding: 16px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 14px;
      min-height: 220px;
      justify-content: center;
    }
    .math-view {
      font-size: 28px;
      min-height: 48px;
      display: flex;
      align-items: center;
      justify-content: center;
    }
    .badge-row {
      display: flex;
      gap: 8px;
      align-items: center;
    }
    .badge {
      padding: 4px 12px;
      border-radius: 20px;
      font-size: 14px;
      font-weight: 600;
      display: inline-flex;
      align-items: center;
      gap: 4px;
    }
    .badge.valid {
      background: #ecfdf5;
      color: #059669;
      border: 1px solid #a7f3d0;
    }
    .badge.invalid {
      background: #fef2f2;
      color: #dc2626;
      border: 1px solid #fecaca;
    }
    .details-grid {
      display: grid;
      grid-template-columns: repeat(auto-fit, minmax(130px, 1fr));
      gap: 8px;
      width: 100%;
      max-width: 440px;
    }
    .detail-card {
      background: #f8f9fa;
      border-radius: 8px;
      padding: 8px 10px;
      text-align: center;
      font-size: 13px;
      display: flex;
      flex-direction: column;
      gap: 2px;
    }
    .detail-label {
      color: #6b7280;
      font-size: 11px;
      text-transform: uppercase;
      letter-spacing: 0.5px;
    }
    .detail-val {
      font-weight: 600;
      color: #1e2945;
    }
    .rule-box {
      font-size: 13px;
      padding: 6px 12px;
      border-radius: 6px;
      text-align: center;
      width: 100%;
      max-width: 440px;
    }
    .rule-box.valid {
      background: #f0fdf4;
      color: #166534;
    }
    .rule-box.invalid {
      background: #fff1f2;
      color: #991b1b;
    }
  </style>
</head>
<body>

  <div class="tabs" id="tabGroup"></div>

  <div class="display-panel">
    <div class="math-view" id="mathTarget"></div>
    <div class="badge-row">
      <div class="badge" id="statusBadge"></div>
    </div>
    <div class="details-grid" id="detailsGrid"></div>
    <div class="rule-box" id="ruleMsg"></div>
  </div>

  <script>
    const examples = [
      {
        id: 'ex1',
        label: '4x³ - 2x + 7',
        tex: '4x^3 - 2x + 7',
        poly: true,
        degree: '3 (Cubic)',
        terms: '3 (Trinomial)',
        rule: '✓ Whole exponents (3, 1, 0)'
      },
      {
        id: 'ex2',
        label: '3x⁻² + 5',
        tex: '3x^{-2} + 5',
        poly: false,
        degree: 'Undefined',
        terms: '2 terms',
        rule: '✗ Exponent -2 is negative'
      },
      {
        id: 'ex3',
        label: '√x + 4',
        tex: '\\sqrt{x} + 4 = x^{\\frac{1}{2}} + 4',
        poly: false,
        degree: 'Undefined',
        terms: '2 terms',
        rule: '✗ Exponent 1/2 is not an integer'
      },
      {
        id: 'ex4',
        label: '5/x² - 1',
        tex: '\\frac{5}{x^2} - 1 = 5x^{-2} - 1',
        poly: false,
        degree: 'Undefined',
        terms: '2 terms',
        rule: '✗ Variable in denominator'
      },
      {
        id: 'ex5',
        label: '2x⁴ - ½x² + π',
        tex: '2x^4 - \\frac{1}{2}x^2 + \\pi',
        poly: true,
        degree: '4 (Quartic)',
        terms: '3 (Trinomial)',
        rule: '✓ Coefficients can be fractions/constants'
      }
    ];

    let currentIndex = 0;

    function renderTabs() {
      const group = document.getElementById('tabGroup');
      group.innerHTML = '';
      examples.forEach((item, idx) => {
        const btn = document.createElement('button');
        btn.className = 'tab-btn' + (idx === currentIndex ? ' active' : '');
        btn.innerText = item.label;
        btn.onclick = () => {
          currentIndex = idx;
          updateView();
          renderTabs();
        };
        group.appendChild(btn);
      });
    }

    function updateView() {
      const data = examples[currentIndex];
      const mathTarget = document.getElementById('mathTarget');
      const statusBadge = document.getElementById('statusBadge');
      const detailsGrid = document.getElementById('detailsGrid');
      const ruleMsg = document.getElementById('ruleMsg');

      if (window.katex) {
        katex.render(data.tex, mathTarget, { throwOnError: false });
      } else {
        mathTarget.textContent = data.label;
      }

      if (data.poly) {
        statusBadge.className = 'badge valid';
        statusBadge.innerHTML = '✓ Polynomial';
        ruleMsg.className = 'rule-box valid';
      } else {
        statusBadge.className = 'badge invalid';
        statusBadge.innerHTML = '✗ Not a Polynomial';
        ruleMsg.className = 'rule-box invalid';
      }

      detailsGrid.innerHTML = `
        <div class="detail-card">
          <span class="detail-label">Degree</span>
          <span class="detail-val">${data.degree}</span>
        </div>
        <div class="detail-card">
          <span class="detail-label">Terms</span>
          <span class="detail-val">${data.terms}</span>
        </div>
      `;

      ruleMsg.textContent = data.rule;
    }

    renderTabs();
    updateView();
  </script>
</body>
</html>

To easily read and compare polynomials, mathematicians arrange them in a strict order.

A polynomial is in **standard form** when its terms are written in descending order of exponents, starting with the highest power on the left. The highest exponent in the expression is the **degree** of the polynomial, and the number attached to that highest power is the **leading coefficient**.

<!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, sans-serif;
      background: #f8f9fa;
      color: #1e2945;
      padding: 16px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
    }

    .container {
      width: 100%;
      max-width: 540px;
      display: flex;
      flex-direction: column;
      gap: 16px;
      align-items: center;
    }

    /* Tabs */
    .tabs {
      display: flex;
      gap: 8px;
      width: 100%;
      justify-content: center;
      flex-wrap: wrap;
    }

    .tab-btn {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 20px;
      padding: 6px 14px;
      font-size: 13px;
      font-weight: 600;
      color: #64748b;
      cursor: pointer;
      box-shadow: 0 1px 2px rgba(0,0,0,0.04);
      transition: all 0.2s ease;
    }

    .tab-btn.active {
      background: #22b7ff;
      color: #ffffff;
      border-color: #22b7ff;
      box-shadow: 0 2px 6px rgba(34,183,255,0.3);
    }

    /* Main Display Area */
    .display-box {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 16px;
      width: 100%;
      padding: 24px 16px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 20px;
      box-shadow: 0 4px 12px rgba(0,0,0,0.03);
    }

    /* Equation with dissectible parts */
    .eq-row {
      display: flex;
      align-items: center;
      justify-content: center;
      font-size: 26px;
      font-family: 'KaTeX_Main', serif;
      gap: 4px;
      user-select: none;
      flex-wrap: wrap;
    }

    .part {
      padding: 4px 6px;
      border-radius: 8px;
      transition: all 0.25s ease;
      display: inline-flex;
      align-items: baseline;
      position: relative;
      cursor: pointer;
    }

    .highlight-lead {
      background: #e0f2fe;
      box-shadow: 0 0 0 2px #0284c7;
      color: #0369a1;
    }

    .highlight-deg {
      background: #fef3c7;
      box-shadow: 0 0 0 2px #d97706;
      color: #b45309;
    }

    .highlight-const {
      background: #f3e8ff;
      box-shadow: 0 0 0 2px #9333ea;
      color: #7e22ce;
    }

    .highlight-term {
      background: #ecfdf5;
      box-shadow: 0 0 0 2px #059669;
      color: #047857;
    }

    /* Focus Badges */
    .badges-grid {
      display: grid;
      grid-template-columns: repeat(2, 1fr);
      gap: 10px;
      width: 100%;
    }

    .badge-card {
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 12px;
      padding: 10px 12px;
      display: flex;
      flex-direction: column;
      gap: 2px;
      cursor: pointer;
      transition: all 0.2s ease;
    }

    .badge-card:hover {
      border-color: #cbd5e1;
    }

    .badge-card.active-lead {
      background: #f0f9ff;
      border-color: #0284c7;
      box-shadow: 0 2px 8px rgba(2, 132, 199, 0.15);
    }

    .badge-card.active-deg {
      background: #fffbeb;
      border-color: #d97706;
      box-shadow: 0 2px 8px rgba(217, 119, 6, 0.15);
    }

    .badge-card.active-const {
      background: #faf5ff;
      border-color: #9333ea;
      box-shadow: 0 2px 8px rgba(147, 51, 234, 0.15);
    }

    .badge-card.active-terms {
      background: #f0fdf4;
      border-color: #059669;
      box-shadow: 0 2px 8px rgba(5, 150, 105, 0.15);
    }

    .badge-title {
      font-size: 11px;
      font-weight: 700;
      text-transform: uppercase;
      letter-spacing: 0.5px;
      color: #64748b;
    }

    .badge-val {
      font-size: 16px;
      font-weight: 700;
      color: #1e2945;
    }

    /* Exponent order bar */
    .order-indicator {
      display: flex;
      align-items: center;
      gap: 8px;
      font-size: 12px;
      font-weight: 600;
      color: #64748b;
      background: #f1f5f9;
      padding: 4px 12px;
      border-radius: 12px;
    }

    .order-arrow {
      color: #22b7ff;
      font-weight: bold;
    }
  </style>
</head>
<body>

<div class="container">
  
  <div class="tabs" id="polyTabs">
    <button class="tab-btn active" data-idx="0">Example 1</button>
    <button class="tab-btn" data-idx="1">Example 2</button>
    <button class="tab-btn" data-idx="2">Example 3</button>
  </div>

  
  <div class="display-box">
    <div class="order-indicator">
      <span>Standard Form</span>
      <span class="order-arrow">Powers: 3 → 2 → 1 → 0</span>
    </div>

    
    <div class="eq-row" id="eqContainer"></div>

    
    <div class="badges-grid">
      <div class="badge-card" id="btnLead" data-mode="lead">
        <span class="badge-title">Leading Coeff.</span>
        <span class="badge-val" id="valLead">5</span>
      </div>
      <div class="badge-card" id="btnDeg" data-mode="deg">
        <span class="badge-title">Degree</span>
        <span class="badge-val" id="valDeg">3</span>
      </div>
      <div class="badge-card" id="btnTerms" data-mode="terms">
        <span class="badge-title">Total Terms</span>
        <span class="badge-val" id="valTerms">4</span>
      </div>
      <div class="badge-card" id="btnConst" data-mode="const">
        <span class="badge-title">Constant Term</span>
        <span class="badge-val" id="valConst">-4</span>
      </div>
    </div>
  </div>
</div>

<script>
  const data = [
    {
      powersOrder: "Powers: 3 → 2 → 1 → 0",
      lead: "5",
      deg: "3",
      termsCount: "4",
      constVal: "-4",
      html: `
        <span class="part term-0" data-part="term"><span class="part" data-part="lead">5</span>x<sup class="part" data-part="deg">3</sup></span>
        <span>&minus;</span>
        <span class="part term-1" data-part="term">2x<sup>2</sup></span>
        <span>+</span>
        <span class="part term-2" data-part="term">7x</span>
        <span>&minus;</span>
        <span class="part term-3" data-part="const">4</span>
      `
    },
    {
      powersOrder: "Powers: 4 → 2 → 0",
      lead: "-3",
      deg: "4",
      termsCount: "3",
      constVal: "+8",
      html: `
        <span class="part term-0" data-part="term"><span class="part" data-part="lead">&minus;3</span>x<sup class="part" data-part="deg">4</sup></span>
        <span>+</span>
        <span class="part term-1" data-part="term">x<sup>2</sup></span>
        <span>+</span>
        <span class="part term-2" data-part="const">8</span>
      `
    },
    {
      powersOrder: "Powers: 2 → 1",
      lead: "6",
      deg: "2",
      termsCount: "2",
      constVal: "0",
      html: `
        <span class="part term-0" data-part="term"><span class="part" data-part="lead">6</span>x<sup class="part" data-part="deg">2</sup></span>
        <span>&minus;</span>
        <span class="part term-1" data-part="term">9x</span>
      `
    }
  ];

  let currentIdx = 0;
  let activeMode = null;

  const eqContainer = document.getElementById('eqContainer');
  const orderIndicator = document.querySelector('.order-arrow');
  const valLead = document.getElementById('valLead');
  const valDeg = document.getElementById('valDeg');
  const valTerms = document.getElementById('valTerms');
  const valConst = document.getElementById('valConst');

  const btnLead = document.getElementById('btnLead');
  const btnDeg = document.getElementById('btnDeg');
  const btnTerms = document.getElementById('btnTerms');
  const btnConst = document.getElementById('btnConst');
  const badges = [btnLead, btnDeg, btnTerms, btnConst];

  function render(idx) {
    const d = data[idx];
    orderIndicator.textContent = d.powersOrder;
    valLead.textContent = d.lead;
    valDeg.textContent = d.deg;
    valTerms.textContent = d.termsCount;
    valConst.textContent = d.constVal;
    eqContainer.innerHTML = d.html;
    applyHighlight(activeMode);
  }

  function applyHighlight(mode) {
    activeMode = mode;
    
    // Clear highlights on badges
    badges.forEach(b => {
      b.className = 'badge-card';
    });

    // Clear highlights on equation
    const parts = eqContainer.querySelectorAll('.part');
    parts.forEach(p => {
      p.classList.remove('highlight-lead', 'highlight-deg', 'highlight-const', 'highlight-term');
    });

    if (!mode) return;

    // Highlight active badge
    if (mode === 'lead') {
      btnLead.classList.add('active-lead');
      eqContainer.querySelectorAll('[data-part="lead"]').forEach(el => el.classList.add('highlight-lead'));
    } else if (mode === 'deg') {
      btnDeg.classList.add('active-deg');
      eqContainer.querySelectorAll('[data-part="deg"]').forEach(el => el.classList.add('highlight-deg'));
    } else if (mode === 'const') {
      btnConst.classList.add('active-const');
      eqContainer.querySelectorAll('[data-part="const"]').forEach(el => el.classList.add('highlight-const'));
    } else if (mode === 'terms') {
      btnTerms.classList.add('active-terms');
      eqContainer.querySelectorAll('[data-part="term"], [data-part="const"]').forEach(el => el.classList.add('highlight-term'));
    }
  }

  // Preset tab clicks
  document.querySelectorAll('.tab-btn').forEach(btn => {
    btn.addEventListener('click', () => {
      document.querySelectorAll('.tab-btn').forEach(b => b.classList.remove('active'));
      btn.classList.add('active');
      currentIdx = parseInt(btn.dataset.idx, 10);
      render(currentIdx);
    });
  });

  // Badge card clicks
  badges.forEach(b => {
    b.addEventListener('click', () => {
      const mode = b.dataset.mode;
      if (activeMode === mode) {
        applyHighlight(null);
      } else {
        applyHighlight(mode);
      }
    });
  });

  // Delegate click on equation parts
  eqContainer.addEventListener('click', (e) => {
    const target = e.target.closest('.part');
    if (!target) return;
    const partType = target.dataset.part;
    if (partType) {
      applyHighlight(activeMode === partType ? null : partType);
    }
  });

  // Initial render with Leading Coefficient highlighted
  render(0);
  applyHighlight('lead');
</script>

</body>
</html>

Once a polynomial is organized in standard form, how do we give it an official name?

Just like a person has a first and last name, every polynomial has a two-part name: its first name comes from its **degree**, and its last name comes from its **number of terms**.

<!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;
      font-family: system-ui, -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, sans-serif;
      user-select: none;
      -webkit-user-select: none;
    }

    body {
      background: transparent;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      padding: 12px 8px;
      min-height: 100%;
    }

    .nav-row {
      display: flex;
      flex-wrap: wrap;
      gap: 6px;
      justify-content: center;
      width: 100%;
      max-width: 480px;
      margin-bottom: 16px;
    }

    .poly-btn {
      background: #ffffff;
      border: 2px solid #e2e8f0;
      border-radius: 20px;
      padding: 6px 12px;
      font-size: 14px;
      font-weight: 600;
      color: #64748b;
      cursor: pointer;
      transition: all 0.2s ease;
    }

    .poly-btn.active {
      background: #0284c7;
      border-color: #0284c7;
      color: #ffffff;
      box-shadow: 0 2px 8px rgba(2, 132, 199, 0.25);
    }

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

    .expr-container {
      display: flex;
      align-items: center;
      justify-content: center;
      flex-wrap: wrap;
      gap: 8px;
      padding: 14px 12px;
      background: #ffffff;
      border-radius: 16px;
      border: 1px solid #e2e8f0;
      box-shadow: 0 2px 8px rgba(0, 0, 0, 0.04);
      min-height: 72px;
      width: 100%;
    }

    .term-card {
      position: relative;
      display: inline-flex;
      align-items: baseline;
      padding: 6px 10px;
      background: #f0fdf4;
      border: 1.5px dashed #22c55e;
      border-radius: 8px;
      font-size: 24px;
      font-weight: 700;
      color: #15803d;
      transition: transform 0.2s ease;
    }

    .exp-highlight {
      display: inline-block;
      background: #e0f2fe;
      color: #0369a1;
      border: 1.5px solid #0284c7;
      border-radius: 6px;
      padding: 0 4px;
      margin-left: 2px;
      transform: translateY(-8px);
      font-size: 16px;
      line-height: 1.2;
    }

    .term-num {
      position: absolute;
      top: -9px;
      right: -7px;
      background: #16a34a;
      color: #ffffff;
      font-size: 10px;
      font-weight: 800;
      width: 17px;
      height: 17px;
      border-radius: 50%;
      display: flex;
      align-items: center;
      justify-content: center;
    }

    .arrows-grid {
      display: grid;
      grid-template-columns: 1fr 1fr;
      width: 100%;
      gap: 12px;
    }

    .name-box {
      background: #ffffff;
      border-radius: 12px;
      padding: 10px 8px;
      text-align: center;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 4px;
      border: 1.5px solid #e2e8f0;
    }

    .name-box.first {
      border-top: 3px solid #0284c7;
    }

    .name-box.last {
      border-top: 3px solid #16a34a;
    }

    .source-label {
      font-size: 11px;
      font-weight: 700;
      text-transform: uppercase;
      letter-spacing: 0.5px;
    }

    .first .source-label {
      color: #0284c7;
    }

    .last .source-label {
      color: #16a34a;
    }

    .classified-val {
      font-size: 16px;
      font-weight: 800;
      color: #0f172a;
    }

    .full-name-banner {
      width: 100%;
      background: #f8fafc;
      border: 2px solid #cbd5e1;
      border-radius: 12px;
      padding: 10px 14px;
      display: flex;
      justify-content: center;
      align-items: center;
      gap: 8px;
      font-size: 18px;
      font-weight: 800;
    }

    .tag-first {
      color: #0284c7;
    }

    .tag-last {
      color: #16a34a;
    }
  </style>
</head>
<body>

  <div class="nav-row" id="nav">
    <button class="poly-btn active" onclick="show(0)">5x³ - 2x + 7</button>
    <button class="poly-btn" onclick="show(1)">4x² - 9</button>
    <button class="poly-btn" onclick="show(2)">8x</button>
    <button class="poly-btn" onclick="show(3)">3x⁴ + x² - 5</button>
  </div>

  <div class="workspace">
    <div class="expr-container" id="exprBox"></div>

    <div class="arrows-grid">
      <div class="name-box first">
        <span class="source-label">Degree (First Name)</span>
        <span class="classified-val" id="firstNameVal">Cubic</span>
      </div>
      <div class="name-box last">
        <span class="source-label">Terms (Last Name)</span>
        <span class="classified-val" id="lastNameVal">Trinomial</span>
      </div>
    </div>

    <div class="full-name-banner">
      <span class="tag-first" id="bannerFirst">Cubic</span>
      <span class="tag-last" id="bannerLast">Trinomial</span>
    </div>
  </div>

  <script>
    const data = [
      {
        terms: [
          { html: '5x<span class="exp-highlight">3</span>', num: 1 },
          { html: '- 2x', num: 2 },
          { html: '+ 7', num: 3 }
        ],
        degreeName: 'Cubic (3°)',
        termsName: 'Trinomial (3)',
        firstName: 'Cubic',
        lastName: 'Trinomial'
      },
      {
        terms: [
          { html: '4x<span class="exp-highlight">2</span>', num: 1 },
          { html: '- 9', num: 2 }
        ],
        degreeName: 'Quadratic (2°)',
        termsName: 'Binomial (2)',
        firstName: 'Quadratic',
        lastName: 'Binomial'
      },
      {
        terms: [
          { html: '8x<span class="exp-highlight">1</span>', num: 1 }
        ],
        degreeName: 'Linear (1°)',
        termsName: 'Monomial (1)',
        firstName: 'Linear',
        lastName: 'Monomial'
      },
      {
        terms: [
          { html: '3x<span class="exp-highlight">4</span>', num: 1 },
          { html: '+ x²', num: 2 },
          { html: '- 5', num: 3 }
        ],
        degreeName: 'Quartic (4°)',
        termsName: 'Trinomial (3)',
        firstName: 'Quartic',
        lastName: 'Trinomial'
      }
    ];

    function show(index) {
      const btns = document.querySelectorAll('.poly-btn');
      btns.forEach((b, i) => b.classList.toggle('active', i === index));

      const item = data[index];
      const exprBox = document.getElementById('exprBox');
      
      exprBox.innerHTML = item.terms.map(t => `
        <div class="term-card">
          ${t.html}
          <div class="term-num">${t.num}</div>
        </div>
      `).join('');

      document.getElementById('firstNameVal').textContent = item.degreeName;
      document.getElementById('lastNameVal').textContent = item.termsName;
      document.getElementById('bannerFirst').textContent = item.firstName;
      document.getElementById('bannerLast').textContent = item.lastName;
    }

    show(0);
  </script>
</body>
</html>

What happens when an expression has terms that can be combined before naming?

Let's classify the expression $6x - 4x^3 + 7 + 9x^3 - x^2$. First, combine any **like terms** — terms that share the exact same variable and exponent.

Combining $-4x^3$ and $+9x^3$ gives $+5x^3$. Ordering from highest to lowest exponent gives the standard form: $$5x^3 - x^2 + 6x + 7$$

The highest exponent is $3$, making it **cubic**. It contains $4$ separate terms, so its complete classification is a **cubic polynomial of 4 terms** with a leading coefficient of $5$.

A common mistake is reading the leading coefficient from the very first term before sorting into standard form — always reorder by exponent first, then identify your leader!

math

Introduction to Polynomials: Degree, Terms, and Classification. Identify the degree, terms, leading coefficient, and name of a polynomial.

The Parts of Polynomial Expressions & Polynomials Intro

Polynomial Vocabulary: Degree, Terms, and Leading Coefficients

OpenStax Algebra: Identifying Degree, Leading Coefficient, and Classification of Polynomials

When introducing polynomials to 9th-10th grade students, anchor the concept by explicitly distinguishing between terms that qualify as polynomials versus expressions containing radicals or variable denominators (connecting with exponent properties in HSN.RN.A.2). Use high-contrast color coding to associate each coefficient with its preceding sign. Emphasize that standard form is not just a stylistic convention, but a vital prerequisite for synthetic/long division, end-behavior analysis, and polynomial factoring.

Introduction to Polynomials: Degree, Terms, and Classification

Anatomy of a Term

What Makes a Polynomial?

Standard Form and Degree