Polynomial Functions
Complete Notes & Formulae for Twelfth Grade (Precalculus)
1. Find the Roots of Factored Polynomials
Zero Product Property:
If the product of factors equals zero, then at least one factor must equal zero
\[ \text{If } A \cdot B = 0, \text{ then } A = 0 \text{ or } B = 0 \]
Method:
1. Set each factor equal to zero
2. Solve each equation
3. The solutions are the roots (zeros) of the polynomial
Example:
Find roots of: \( f(x) = (x - 2)(x + 3)(x - 5) \)
Set each factor to zero:
\( x - 2 = 0 \) → \( x = 2 \)
\( x + 3 = 0 \) → \( x = -3 \)
\( x - 5 = 0 \) → \( x = 5 \)
Roots: x = -3, 2, 5
2. Write a Polynomial from Its Roots
Formula:
If \( r_1, r_2, r_3, \ldots, r_n \) are the roots, then:
\[ P(x) = a(x - r_1)(x - r_2)(x - r_3) \cdots (x - r_n) \]
where \( a \) is any non-zero constant (typically 1 or the leading coefficient)
Example:
Write a polynomial with roots: -2, 1, and 4
\( P(x) = (x - (-2))(x - 1)(x - 4) \)
\( P(x) = (x + 2)(x - 1)(x - 4) \)
Expanded: \( P(x) = x^3 - 3x^2 - 6x + 8 \)
3. Rational Root Theorem
Statement:
For polynomial \( P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 \) with integer coefficients:
\[ \text{Possible rational roots: } \frac{p}{q} \]
where:
• \( p \) = factor of the constant term \( a_0 \)
• \( q \) = factor of the leading coefficient \( a_n \)
• List includes both positive and negative values: \( \pm\frac{p}{q} \)
Example:
Find possible rational roots: \( 2x^3 - 5x^2 + 3x - 6 = 0 \)
Constant term: \( a_0 = -6 \) → Factors: ±1, ±2, ±3, ±6
Leading coefficient: \( a_n = 2 \) → Factors: ±1, ±2
Possible rational roots: \( \pm 1, \pm 2, \pm 3, \pm 6, \pm \frac{1}{2}, \pm \frac{3}{2} \)
4. Complex Conjugate Theorem
Theorem:
If a polynomial has real coefficients and \( a + bi \) (where \( b \neq 0 \)) is a root, then its complex conjugate \( a - bi \) is also a root
\[ \text{If } (a + bi) \text{ is a root, then } (a - bi) \text{ is also a root} \]
Example:
If \( 3 + 4i \) is a root, then \( 3 - 4i \) must also be a root
Complex roots always come in conjugate pairs for polynomials with real coefficients
5. Conjugate Root Theorems
Includes both:
1. Complex Conjugate Theorem:
Complex roots come in conjugate pairs: \( a \pm bi \)
2. Irrational Conjugate Theorem:
If a polynomial with rational coefficients has \( a + \sqrt{b} \) as a root (where \( \sqrt{b} \) is irrational), then \( a - \sqrt{b} \) is also a root
Examples:
• If \( 2 + \sqrt{3} \) is a root, then \( 2 - \sqrt{3} \) is also a root
• If \( 1 + 2i \) is a root, then \( 1 - 2i \) is also a root
6. Descartes' Rule of Signs
Statement:
For Positive Real Roots:
Count the number of sign changes in \( P(x) \). The number of positive real roots is either equal to this count or less by an even number.
For Negative Real Roots:
Count the number of sign changes in \( P(-x) \). The number of negative real roots is either equal to this count or less by an even number.
Example:
Analyze: \( P(x) = x^4 - 3x^3 + 2x^2 + x - 5 \)
For positive roots (P(x)):
Signs: + − + + −
Sign changes: 3 (+ to −, − to +, + to −)
Possible positive roots: 3 or 1
For negative roots (P(-x)):
\( P(-x) = x^4 + 3x^3 + 2x^2 - x - 5 \)
Signs: + + + − −
Sign changes: 1 (+ to −)
Possible negative roots: 1
7. Fundamental Theorem of Algebra
Theorem:
Every polynomial of degree \( n \geq 1 \) with complex coefficients has exactly \( n \) complex roots (counting multiplicities)
\[ \text{Degree } n \text{ polynomial} \rightarrow n \text{ roots (including complex)} \]
Key Points:
• A cubic (degree 3) has exactly 3 roots
• A quartic (degree 4) has exactly 4 roots
• Roots can be real or complex
• Complex roots come in conjugate pairs (for real coefficients)
• Roots can repeat (multiplicity > 1)
8. Match Polynomials and Graphs Using End Behavior
End Behavior Rules:
End behavior depends on the degree and leading coefficient:
| Degree | Leading Coefficient | Left End | Right End |
|---|---|---|---|
| Even | Positive (+) | ↑ (up) | ↑ (up) |
| Even | Negative (−) | ↓ (down) | ↓ (down) |
| Odd | Positive (+) | ↓ (down) | ↑ (up) |
| Odd | Negative (−) | ↑ (up) | ↓ (down) |
9. Match Polynomials and Graphs Using Zeros
Key Concepts:
Zeros (x-intercepts):
Points where the graph crosses or touches the x-axis
Multiplicity:
• Odd multiplicity (1, 3, 5...): Graph crosses x-axis
• Even multiplicity (2, 4, 6...): Graph touches x-axis and bounces back
Example:
\( f(x) = (x - 2)^2(x + 1) \)
• Zero at x = 2 (multiplicity 2): touches and bounces
• Zero at x = -1 (multiplicity 1): crosses
10. Domain and Range of Polynomials
General Rules:
Domain:
ALL polynomial functions have domain: \( (-\infty, \infty) \)
Polynomials are defined for all real numbers
Range:
• Even degree, positive leading coefficient: \( [y_{min}, \infty) \)
• Even degree, negative leading coefficient: \( (-\infty, y_{max}] \)
• Odd degree (any leading coefficient): \( (-\infty, \infty) \)
11. Even and Odd Functions
Definitions:
Even Function:
\[ f(-x) = f(x) \]
• Symmetric about the y-axis
• Contains only even powers of x
• Example: \( f(x) = x^4 - 2x^2 + 1 \)
Odd Function:
\[ f(-x) = -f(x) \]
• Symmetric about the origin (rotational symmetry)
• Contains only odd powers of x
• Example: \( f(x) = x^3 - 5x \)
Neither Even nor Odd:
• Contains both even and odd powers
• Example: \( f(x) = x^3 + x^2 + 1 \)
12. Quick Reference Summary
Key Theorems & Rules:
Rational Root Theorem: \( \frac{p}{q} \) where p | \( a_0 \), q | \( a_n \)
Complex Conjugate: If \( a + bi \) is a root, then \( a - bi \) is too
Fundamental Theorem: Degree n → n roots (counting multiplicity)
Descartes' Rule: Sign changes → max positive/negative roots
Domain of Polynomials: Always \( (-\infty, \infty) \)
Even Function: \( f(-x) = f(x) \) (y-axis symmetry)
Odd Function: \( f(-x) = -f(x) \) (origin symmetry)
📚 Study Tips
✓ Use Rational Root Theorem to find possible rational roots, then test
✓ Complex roots ALWAYS come in conjugate pairs for real coefficients
✓ Descartes' Rule gives maximum possible roots, not exact number
✓ Degree tells you total number of roots (Fundamental Theorem)
✓ Even multiplicity = bounce; Odd multiplicity = cross