Understanding Polynomials
A polynomial is an expression with one or more terms, each involving a variable raised to a non-negative integer power. The degree is the highest power.
P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀
Example: P(x) = 2x³ − 4x² + x − 5
Example: P(x) = 2x³ − 4x² + x − 5
Types of Polynomials
Monomial
1 term: 5x²
Binomial
2 terms: x + 3
Trinomial
3 terms: x²+x+1
Polynomial
4+ terms
Polynomial Degree Reference
| Degree | Name | General Form | Example |
|---|---|---|---|
| 0 | Constant | a | 7 |
| 1 | Linear | ax + b | 3x + 5 |
| 2 | Quadratic | ax² + bx + c | 2x² − 3x + 1 |
| 3 | Cubic | ax³ + bx² + cx + d | x³ − 6x² + 11x − 6 |
| 4 | Quartic | ax⁴ + ... | x⁴ − 5x² + 4 |
| n | nth degree | aₙxⁿ + ... + a₀ | — |
Evaluating Polynomials — Worked Examples
Example 1: Quadratic
P(x) = 2x² − 3x + 5 at x = 4
P(4) = 2(16) − 3(4) + 5 = 32 − 12 + 5 = 25
P(4) = 2(16) − 3(4) + 5 = 32 − 12 + 5 = 25
Example 2: Cubic
P(x) = x³ − 2x² + x − 1 at x = 3
P(3) = 27 − 18 + 3 − 1 = 11
P(3) = 27 − 18 + 3 − 1 = 11
Horner's Method — Efficient Evaluation
Horner's method reduces the number of multiplications needed to evaluate a polynomial:
P(x) = aₙxⁿ + ... + a₁x + a₀
Rewrite: P(x) = (...((aₙx + aₙ₋₁)x + aₙ₋₂)x + ... + a₁)x + a₀
Example: x³−2x²+x−1 at x=3
Start: 1
Step 1: 1×3 + (−2) = 1
Step 2: 1×3 + 1 = 4
Step 3: 4×3 + (−1) = 11 ✓
Rewrite: P(x) = (...((aₙx + aₙ₋₁)x + aₙ₋₂)x + ... + a₁)x + a₀
Example: x³−2x²+x−1 at x=3
Start: 1
Step 1: 1×3 + (−2) = 1
Step 2: 1×3 + 1 = 4
Step 3: 4×3 + (−1) = 11 ✓
Real-World Applications
- Engineering curves: Polynomial interpolation fits curves through data points
- Computer graphics: Bézier curves use polynomials for smooth shape rendering
- Physics: Taylor series approximations use polynomials to represent functions like sin(x), eˣ
- Economics: Cost, revenue, and profit functions are often modelled as polynomials
- Signal processing: Filter design uses polynomial equations