📈 Algebra: Polynomial Functions

Understanding higher-degree functions and their behaviors

🎯 Learning Objectives

  • Identify and classify polynomial functions by degree
  • Understand end behavior and turning points
  • Factor polynomials using various techniques
  • Graph polynomial functions and identify key features
  • Apply polynomial division and the remainder theorem
  • Solve polynomial equations

1. Understanding Polynomials

🔢 Polynomial Functions

A polynomial function is a function of the form:

f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0

Where n is a non-negative integer and aₙ ≠ 0.

Polynomial Classification by Degree:

Degree 0

Constant: f(x) = 5

Horizontal line

Degree 1

Linear: f(x) = 2x + 3

Straight line

Degree 2

Quadratic: f(x) = x² + 2x + 1

Parabola

Degree 3

Cubic: f(x) = x³ - x

S-shaped curve

🧱 Polynomial Builder

Build and explore polynomial functions by adjusting coefficients:

x³
x²
x
(constant)
f(x) = x³ - 2x + 1

Degree: 3

Leading coefficient: 1

End behavior: ↑↑ (up and up)

2. End Behavior of Polynomials

📊 Understanding End Behavior

The end behavior of a polynomial is determined by its degree and leading coefficient.

Even degree Positive leading coefficient Both ends up: ↑↑
Even degree Negative leading coefficient Both ends down: ↓↓
Odd degree Positive leading coefficient Left down, right up: ↓↑
Odd degree Negative leading coefficient Left up, right down: ↑↓
📈 End Behavior Practice

f(x) = 2x³ - x² + 3

Degree: (odd/even?)

Leading coefficient: (positive/negative?)

End behavior:

g(x) = -x⁴ + 2x² - 1

Degree: (odd/even?)

Leading coefficient: (positive/negative?)

End behavior:

3. Factoring Polynomials

🔧 Factoring Techniques

Factoring polynomials means writing them as a product of simpler polynomials.

Common Factoring Methods:

  • Greatest Common Factor (GCF): Factor out the common terms
  • Difference of Squares: a² - b² = (a - b)(a + b)
  • Perfect Square Trinomials: a² ± 2ab + b² = (a ± b)²
  • Grouping: Group terms and factor by groups
  • Sum/Difference of Cubes: a³ ± b³ = (a ± b)(a² ∓ ab + b²)
🧩 Polynomial Factoring Practice

Factor the polynomial: x³ - 8

This is a:

Factored form:

📝 Factor These Polynomials

1. x² - 9 =

2. x² + 6x + 9 =

3. 2x³ + 16 =

4. Polynomial Division

➗ Long Division of Polynomials

Polynomial division is similar to long division with numbers.

Example: Divide (x² + 5x + 6) ÷ (x + 2)

x + 3

x + 2 ) x² + 5x + 6

-(x² + 2x)

3x + 6

-(3x + 6)

0

Result: x + 3

➗ Polynomial Division Practice

Divide: ÷

📋 Remainder Theorem

When dividing a polynomial f(x) by (x - a), the remainder equals f(a).

For f(x) = x³ - 2x² + 3x - 1, find f(2):

f(2) =

5. Graphing Polynomial Functions

📊 Key Features of Polynomial Graphs

  • Degree: Determines maximum number of turns (degree - 1)
  • Leading coefficient: Affects end behavior
  • y-intercept: Value when x = 0
  • x-intercepts: Roots of the equation f(x) = 0
  • Turning points: Local maximum/minimum values
🔍 Graph Analysis

Analyze the graph of f(x) = x³ - 3x² - x + 3:

Degree:

Leading coefficient:

End behavior:

// End Behavior Checking document.querySelectorAll('.check-behavior').forEach(btn => { btn.addEventListener('click', function() { const set = this.dataset.set; const degInput = document.getElementById(`deg${set}`); const leadInput = document.getElementById(`lead${set}`); const resultSpan = document.getElementById(`behavior${set}-result`); const degree = parseInt(degInput.value); const leading = parseFloat(leadInput.value); let expectedBehavior = ''; if (set === '1') { // 2x³ - x² + 3 expectedBehavior = degree === 3 && leading === 2 ? '↓↑ (left down, right up)' : 'Incorrect'; } else if (set === '2') { // -x⁴ + 2x² - 1 expectedBehavior = degree === 4 && leading === -1 ? '↓↓ (both ends down)' : 'Incorrect'; } resultSpan.innerHTML = expectedBehavior; resultSpan.style.color = expectedBehavior.includes('Incorrect') ? '#c62828' : '#2e7d32'; }); }); // Factoring Practice document.getElementById('check-factoring').addEventListener('click', function() { const type = document.getElementById('factor-type').value; const factored = document.getElementById('factored-form').value.trim(); const resultDiv = document.getElementById('factoring-result'); let correct = false; let message = ''; // For x³ - 8 = (x - 2)(x² + 2x + 4) if (type === 'difference-cubes' && (factored === '(x-2)(x²+2x+4)' || factored === '(x - 2)(x² + 2x + 4)')) { correct = true; message = '🎉 Correct! x³ - 8 = (x - 2)(x² + 2x + 4)'; } else if (type === 'difference-cubes') { message = '❌ Check your factoring. Remember the difference of cubes formula!'; } else { message = '❌ Choose the correct factoring method first.'; } resultDiv.innerHTML = message; resultDiv.className = correct ? 'success-message' : 'error-message'; }); document.getElementById('show-factoring-hint').addEventListener('click', function() { document.getElementById('factoring-hint').style.display = 'block'; }); // Multiple Factoring Problems document.querySelectorAll('.check-factor-prob').forEach(btn => { btn.addEventListener('click', function() { const prob = this.dataset.prob; const expected = this.dataset.answer; const userInput = document.getElementById(`factor${prob}`).value.trim(); const resultSpan = document.getElementById(`factor${prob}-result`); // Normalize input (remove spaces, standardize format) const normalizedUser = userInput.replace(/\s/g, '').toLowerCase(); const normalizedExpected = expected.replace(/\s/g, '').toLowerCase(); if (normalizedUser === normalizedExpected || (prob === '1' && normalizedUser === '(x+3)(x-3)') || (prob === '2' && normalizedUser === '(x+3)^2') || (prob === '3' && normalizedUser === '2(x^3+8)')) { resultSpan.innerHTML = '✅ Correct!'; } else { resultSpan.innerHTML = '❌ Try again!'; } }); }); // Polynomial Division document.getElementById('divide-polynomials').addEventListener('click', function() { const dividend = document.getElementById('dividend').value; const divisor = document.getElementById('divisor').value; const resultDiv = document.getElementById('division-result'); // Simple division for (x² + 5x + 6) ÷ (x + 2) = x + 3 if (dividend.replace(/\s/g, '') === 'x²+5x+6' && divisor.replace(/\s/g, '') === 'x+2') { resultDiv.innerHTML = `
Quotient: x + 3
Remainder: 0
`; } else { resultDiv.innerHTML = '
Division result would be calculated here for more complex polynomials.
'; } }); // Remainder Theorem document.getElementById('check-remainder').addEventListener('click', function() { const answer = parseFloat(document.getElementById('remainder-answer').value); const resultDiv = document.getElementById('remainder-result'); // f(2) = 2³ - 2×2² + 3×2 - 1 = 8 - 2×4 + 6 - 1 = 8 - 8 + 6 - 1 = 5 if (Math.abs(answer - 5) < 0.01) { resultDiv.innerHTML = '
✅ Correct! f(2) = 5
'; } else { resultDiv.innerHTML = '
❌ Incorrect. Calculate 2³ - 2×2² + 3×2 - 1
'; } }); // Graph Analysis document.getElementById('check-analysis').addEventListener('click', function() { const degree = parseInt(document.getElementById('analysis-degree').value); const leading = parseFloat(document.getElementById('analysis-leading').value); const behavior = document.getElementById('analysis-behavior').value; const resultDiv = document.getElementById('analysis-result'); if (degree === 3 && leading === 1 && behavior === 'down-up') { resultDiv.innerHTML = '
🎉 Perfect analysis!
'; } else { resultDiv.innerHTML = '
❌ Check your answers. f(x) = x³ - 3x² - x + 3 has degree 3, leading coefficient 1, and goes from left down to right up.
'; } });
📄 algebra-polynomials.html | 2025-12-26