Polynomial Functions
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What Is a Polynomial Function?
Definition: Polynomial Function
A polynomial function is a function of the form:
f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₂x² + a₁x + a₀
where n is a non-negative integer (the degree), and aₙ, aₙ₋₁, ..., a₀ are real numbers (the coefficients) with aₙ ≠ 0.
Key Vocabulary
- Degree: The highest power of x with a non-zero coefficient
- Leading coefficient: The coefficient of the term with the highest degree (aₙ)
- Leading term: The term with the highest degree (aₙxⁿ)
- Constant term: The term with no variable (a₀)
- Standard form: Terms written in descending order of degree
Classifying Polynomials by Degree
| Degree | Name | Example |
|---|---|---|
| 0 | Constant | f(x) = 5 |
| 1 | Linear | f(x) = 2x + 3 |
| 2 | Quadratic | f(x) = x² - 4x + 1 |
| 3 | Cubic | f(x) = x³ + 2x² - x + 7 |
| 4 | Quartic | f(x) = x⁴ - 3x² + 2 |
| 5 | Quintic | f(x) = x⁵ - x |
End Behavior
The end behavior describes what happens to f(x) as x approaches positive or negative infinity. It depends on:
- The degree (even or odd)
- The sign of the leading coefficient (positive or negative)
End Behavior Rules
| Leading Coeff. | Even Degree | Odd Degree |
|---|---|---|
| Positive (+) | Both ends up: x → ±∞, f(x) → +∞ | Left down, right up: x → -∞, f(x) → -∞; x → +∞, f(x) → +∞ |
| Negative (−) | Both ends down: x → ±∞, f(x) → -∞ | Left up, right down: x → -∞, f(x) → +∞; x → +∞, f(x) → -∞ |
Zeros (Roots) of Polynomials
A zero of a polynomial f(x) is a value c such that f(c) = 0. Zeros are also called roots or x-intercepts.
The Fundamental Theorem of Algebra
Every polynomial of degree n ≥ 1 has exactly n zeros (counting multiplicity and complex zeros).
Multiplicity
If (x - c)^m is a factor of f(x), then c is a zero with multiplicity m.
- Odd multiplicity: Graph crosses the x-axis at this zero
- Even multiplicity: Graph touches but doesn't cross the x-axis at this zero
Finding Zeros: Key Theorems
The Remainder Theorem
If f(x) is divided by (x - c), the remainder equals f(c).
The Factor Theorem
(x - c) is a factor of f(x) if and only if f(c) = 0.
The Rational Zero Theorem
If f(x) has integer coefficients and p/q (in lowest terms) is a rational zero of f, then:
- p is a factor of the constant term (a₀)
- q is a factor of the leading coefficient (aₙ)
Synthetic Division
Synthetic division is a shortcut method for dividing a polynomial by (x - c). It's faster than long division and useful for testing possible zeros.
Examples
Example 1: Identifying Polynomial Properties
Problem: For f(x) = -2x⁴ + 5x³ - x + 8, identify: (a) degree, (b) leading coefficient, (c) constant term, (d) end behavior.
Solution:
(a) Degree: 4 (highest power)
(b) Leading coefficient: -2
(c) Constant term: 8
(d) End behavior: Even degree (4) with negative leading coefficient (-2)
→ Both ends point down: as x → ±∞, f(x) → -∞
Example 2: Using the Rational Zero Theorem
Problem: List all possible rational zeros of f(x) = 2x³ - 3x² - 8x + 12
Solution:
Step 1: Identify factors of constant term (12): ±1, ±2, ±3, ±4, ±6, ±12
Step 2: Identify factors of leading coefficient (2): ±1, ±2
Step 3: Form all possible p/q combinations:
Possible rational zeros: ±1, ±2, ±3, ±4, ±6, ±12, ±1/2, ±3/2
Example 3: Synthetic Division
Problem: Divide f(x) = x³ - 4x² + x + 6 by (x - 2) using synthetic division.
Solution:
2 │ 1 -4 1 6
│ 2 -4 -6
└─────────────────
1 -2 -3 0
The quotient is x² - 2x - 3 with remainder 0.
Since the remainder is 0, (x - 2) is a factor and x = 2 is a zero.
f(x) = (x - 2)(x² - 2x - 3) = (x - 2)(x - 3)(x + 1)
Zeros: x = 2, x = 3, x = -1
Example 4: Analyzing Multiplicity
Problem: For f(x) = (x + 1)²(x - 2)³, find all zeros and describe the graph's behavior at each zero.
Solution:
Zero x = -1 with multiplicity 2 (even) → Graph touches the x-axis
Zero x = 2 with multiplicity 3 (odd) → Graph crosses the x-axis
Total degree: 2 + 3 = 5 (odd) with positive leading coefficient
End behavior: Left down, right up
Example 5: Finding All Zeros
Problem: Find all zeros of f(x) = x³ + 2x² - 5x - 6
Solution:
Step 1: Use Rational Zero Theorem. Possible zeros: ±1, ±2, ±3, ±6
Step 2: Test f(2) = 8 + 8 - 10 - 6 = 0 ✓ So x = 2 is a zero
Step 3: Use synthetic division to factor out (x - 2):
f(x) = (x - 2)(x² + 4x + 3)
Step 4: Factor the quadratic:
x² + 4x + 3 = (x + 1)(x + 3)
Zeros: x = 2, x = -1, x = -3
Practice
Solve these problems. Answers are provided below for self-checking.
1. For f(x) = 3x⁵ - 2x³ + 7x - 1, identify the degree, leading coefficient, and constant term.
2. Describe the end behavior of g(x) = -x³ + 4x² - 2
3. List all possible rational zeros of h(x) = 3x³ + 2x² - x - 2
4. Use synthetic division to divide x³ + 5x² - 2x - 24 by (x + 4)
5. Is x = 3 a zero of f(x) = x³ - 6x² + 11x - 6? Use the Remainder Theorem.
6. Find all zeros of f(x) = x³ - 7x + 6
7. For f(x) = (x - 1)²(x + 2)(x - 3)³, what is the degree? At which zeros does the graph cross the x-axis?
8. Write a polynomial function with zeros at x = -2, x = 1, and x = 4.
9. Find a polynomial of degree 3 with zeros 2, -1, and 3 where f(0) = 12.
10. How many turning points can a polynomial of degree 6 have at most?
Click to reveal answers
- Degree: 5; Leading coefficient: 3; Constant term: -1
- Odd degree (3) with negative leading coefficient: as x → -∞, f(x) → +∞; as x → +∞, f(x) → -∞
- Possible zeros: ±1, ±2, ±1/3, ±2/3
- Quotient: x² + x - 6 with remainder 0; f(x) = (x + 4)(x² + x - 6) = (x + 4)(x + 3)(x - 2)
- f(3) = 27 - 54 + 33 - 6 = 0, so yes, x = 3 is a zero
- x = 1, x = 2, x = -3
- Degree: 2 + 1 + 3 = 6; Crosses at x = -2 (mult. 1) and x = 3 (mult. 3)
- f(x) = (x + 2)(x - 1)(x - 4) = x³ - 3x² - 6x + 8
- f(x) = 2(x - 2)(x + 1)(x - 3) = 2x³ - 8x² + 2x + 12
- At most 5 turning points (degree - 1)
Check Your Understanding
1. How does the degree of a polynomial relate to its maximum number of zeros and turning points?
Show answer
A polynomial of degree n has at most n zeros (exactly n counting multiplicity and complex zeros) and at most n - 1 turning points. For example, a cubic (degree 3) has at most 3 zeros and at most 2 turning points.
2. Why does the Rational Zero Theorem only give "possible" zeros rather than actual zeros?
Show answer
The theorem lists all candidates that could be rational zeros, but not all of them will actually be zeros of the polynomial. You must test each candidate (using substitution or synthetic division) to verify. Also, the polynomial might have irrational or complex zeros not covered by the theorem.
3. Explain how to determine if a graph touches or crosses the x-axis at a zero.
Show answer
Look at the multiplicity of the zero: if the multiplicity is even, the graph touches the x-axis but doesn't cross (it bounces back); if the multiplicity is odd, the graph crosses the x-axis at that point.
4. A polynomial has a positive leading coefficient and its graph goes down on both ends. What can you conclude about its degree?
Show answer
This is impossible! If the leading coefficient is positive: for even degree, both ends go up; for odd degree, left goes down and right goes up. If both ends go down, the leading coefficient must be negative with even degree.
🚀 Next Steps
- Review any concepts that felt challenging
- Move on to the next lesson when ready
- Return to practice problems periodically for review