Lesson 2 • 45 min read
Polynomial Functions
In This Lesson
1Standard Form
f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀
where aₙ ≠ 0 and n is a non-negative integer
Key Terms
- Degree: Highest power of x
- Leading Coefficient: aₙ (coefficient of highest power)
- Constant Term: a₀ (no x)
Example
f(x) = 3x⁴ - 2x² + 5x - 1
- Degree: 4
- Leading coefficient: 3
- Constant term: -1
Types by Degree
Degree 0
Constant
Degree 1
Linear
Degree 2
Quadratic
Degree 3
Cubic
2End Behavior
End behavior describes what happens to f(x) as x approaches positive or negative infinity. It depends on the degree and leading coefficient.
| Degree | Leading Coef | Left End (x→-∞) | Right End (x→+∞) |
|---|---|---|---|
| Even | + (positive) | ↑ (up) | ↑ (up) |
| Even | - (negative) | ↓ (down) | ↓ (down) |
| Odd | + (positive) | ↓ (down) | ↑ (up) |
| Odd | - (negative) | ↑ (up) | ↓ (down) |
Example: f(x) = -2x³ + 5x - 1
Degree: 3 (odd), Leading coefficient: -2 (negative)
End behavior: ↑ left, ↓ right
3Finding Zeros
Zeros (or roots) are x-values where f(x) = 0. These are where the graph crosses or touches the x-axis.
Rational Root Theorem
Possible rational roots = ± (factors of constant)/(factors of leading coefficient)
Example: f(x) = 2x³ - 5x² - x + 6
Factors of 6: ±1, ±2, ±3, ±6
Factors of 2: ±1, ±2
Possible roots: ±1, ±2, ±3, ±6, ±1/2, ±3/2
Multiplicity of Zeros
- Odd multiplicity: Graph crosses x-axis
- Even multiplicity: Graph touches x-axis and bounces back
Example: f(x) = (x - 2)²(x + 1)³
x = 2 has multiplicity 2 (even) → touches
x = -1 has multiplicity 3 (odd) → crosses
4Factor & Remainder Theorems
Factor Theorem
(x - c) is a factor of f(x) if and only if f(c) = 0
If c is a zero, then (x - c) is a factor
Remainder Theorem
When f(x) is divided by (x - c), the remainder is f(c)
Substitute c into f(x) to find the remainder
Example: Is (x - 2) a factor of f(x) = x³ - 4x² + x + 6?
f(2) = (2)³ - 4(2)² + (2) + 6
f(2) = 8 - 16 + 2 + 6
f(2) = 0
Yes! Since f(2) = 0, (x - 2) is a factor
5Synthetic Division
Synthetic division is a quick method to divide a polynomial by (x - c).
Steps
- Write coefficients in a row (include 0 for missing terms)
- Write c (the value from x - c) to the left
- Bring down the first coefficient
- Multiply by c, write under next coefficient
- Add, write result below
- Repeat multiply-add until done
- Last number is remainder; others are quotient coefficients
Example: Divide (x³ - 4x² + x + 6) by (x - 2)
2 | 1 -4 1 6
| 2 -4 -6
----------------------
1 -2 -3 0Quotient: x² - 2x - 3, Remainder: 0
So x³ - 4x² + x + 6 = (x - 2)(x² - 2x - 3)