Polynomial Equations and Functions
📌 What is a Polynomial?
A polynomial is an expression of the form:
\( P(x) = a_nx^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + \cdots + a_1x + a_0 \)
Where \(a_n, a_{n-1}, \ldots, a_0\) are constants, \(a_n \neq 0\), and \(n\) is the degree of the polynomial.
Solving Polynomial Equations
Methods to Solve:
- Factoring: Write the polynomial as a product of linear and/or irreducible factors
- Zero Product Property: If \(AB = 0\), then \(A = 0\) or \(B = 0\)
- Quadratic Formula: For degree 2 polynomials
- Synthetic Division: For testing potential roots
- Graphing: Find where the graph crosses the x-axis
📝 General Factoring Strategy:
- Step 1: Factor out the Greatest Common Factor (GCF)
- Step 2: Determine the number of terms and apply appropriate method
- Step 3: Look for factors that can be factored further
- Step 4: Check by multiplying
🔍 Special Factoring Formulas:
Difference of Squares:
\( a^2 - b^2 = (a-b)(a+b) \)
Perfect Square Trinomials:
\( a^2 + 2ab + b^2 = (a+b)^2 \)
\( a^2 - 2ab + b^2 = (a-b)^2 \)
Sum of Cubes:
\( a^3 + b^3 = (a+b)(a^2-ab+b^2) \)
Difference of Cubes:
\( a^3 - b^3 = (a-b)(a^2+ab+b^2) \)
Roots and Zeros of Polynomials
Finding Roots of Factored Polynomials:
If a polynomial is written in factored form: \( P(x) = a(x-r_1)(x-r_2)\cdots(x-r_n) \)
Then the roots (zeros) are: \( r_1, r_2, \ldots, r_n \)
To find roots: Set each factor equal to zero and solve.
📝 Example:
Find the roots of: \( P(x) = 2(x+3)(x-1)(x-4) \)
Set each factor to zero:
- \( x + 3 = 0 \) → \( x = -3 \)
- \( x - 1 = 0 \) → \( x = 1 \)
- \( x - 4 = 0 \) → \( x = 4 \)
Roots: \( x = -3, 1, 4 \)
Writing a Polynomial from Roots:
If the roots are \( r_1, r_2, \ldots, r_n \), then:
\( P(x) = a(x-r_1)(x-r_2)\cdots(x-r_n) \)
Where \(a\) is any non-zero constant (usually \(a = 1\))
📝 Example:
Write a polynomial with roots: \( x = 2, -1, 5 \)
Solution:
\( P(x) = (x-2)(x+1)(x-5) \)
\( P(x) = x^3 - 6x^2 + 3x + 10 \)
Rational Root Theorem
Theorem Statement:
For a polynomial \( P(x) = a_nx^n + \cdots + a_1x + a_0 \) with integer coefficients:
Any rational root \( \frac{p}{q} \) (in lowest terms) must satisfy:
- \(p\) is a factor of the constant term \(a_0\)
- \(q\) is a factor of the leading coefficient \(a_n\)
Possible Rational Roots = \( \pm \frac{\text{factors of } a_0}{\text{factors of } a_n} \)
📝 Example:
Find possible rational roots of: \( 2x^3 - 5x^2 + 3x + 6 = 0 \)
Step 1: Factors of constant term \(a_0 = 6\): \( \pm 1, \pm 2, \pm 3, \pm 6 \)
Step 2: Factors of leading coefficient \(a_n = 2\): \( \pm 1, \pm 2 \)
Step 3: Possible rational roots: \( \frac{p}{q} \)
\( \pm 1, \pm 2, \pm 3, \pm 6, \pm \frac{1}{2}, \pm \frac{3}{2} \)
Descartes' Rule of Signs
Rule 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 number 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 number or less by an even number.
📝 Example:
Determine possible numbers of positive and negative real roots:
\( P(x) = x^4 - 3x^3 + 2x^2 - 5x + 1 \)
Positive Roots: Count sign changes in \(P(x)\)
+ to − (1), − to + (2), + to − (3), − to + (4) = 4 sign changes
Possible positive roots: 4, 2, or 0
Negative Roots: Find \(P(-x)\)
\( P(-x) = x^4 + 3x^3 + 2x^2 + 5x + 1 \)
0 sign changes
Possible negative roots: 0
Complex Conjugate Theorem
Theorem Statement:
If a polynomial has real coefficients and a complex number \( a + bi \) (where \(b \neq 0\)) is a root, then its complex conjugate \( a - bi \) is also a root.
Key Points:
- Complex roots always come in conjugate pairs
- This applies only to polynomials with real coefficients
- If you know one complex root, you automatically know another
📝 Example:
A polynomial with real coefficients has a root \( 3 + 2i \). What other root must it have?
Answer: \( 3 - 2i \) (the complex conjugate)
Another Example:
Write a polynomial with roots: \( 2, 1+i \)
By the Complex Conjugate Theorem, \( 1-i \) is also a root.
\( P(x) = (x-2)(x-(1+i))(x-(1-i)) \)
\( P(x) = (x-2)[(x-1)-i][(x-1)+i] \)
\( P(x) = (x-2)[(x-1)^2 + 1] \)
\( P(x) = (x-2)(x^2-2x+2) \)
\( P(x) = x^3 - 4x^2 + 6x - 4 \)
Conjugate Root Theorems
Two Types of Conjugate Roots:
1. Complex Conjugate Root Theorem:
If \( a + bi \) is a root, then \( a - bi \) is also a root
2. Irrational Conjugate Root Theorem:
If \( a + \sqrt{b} \) is a root (where \(\sqrt{b}\) is irrational), then \( a - \sqrt{b} \) is also a root
Example: If \( 2 + \sqrt{3} \) is a root, then \( 2 - \sqrt{3} \) is also a root
Fundamental Theorem of Algebra
Theorem Statement:
Every non-constant polynomial with complex coefficients has at least one complex root.
Corollary:
A polynomial of degree \(n\) has exactly \(n\) complex roots (counting multiplicities).
Examples:
- A cubic polynomial (degree 3) has exactly 3 roots
- A quartic polynomial (degree 4) has exactly 4 roots
- Roots can be real or complex
- Complex roots occur in conjugate pairs (for real coefficients)
End Behavior of Polynomials
Leading Coefficient Test:
The end behavior of a polynomial depends on:
- Degree: Even or Odd
- Leading Coefficient: Positive or Negative
End Behavior Rules
1. Even Degree + Positive Leading Coefficient:
As \( x \to -\infty \), \( f(x) \to +\infty \)
As \( x \to +\infty \), \( f(x) \to +\infty \)
Both ends go UP ↑↑
2. Even Degree + Negative Leading Coefficient:
As \( x \to -\infty \), \( f(x) \to -\infty \)
As \( x \to +\infty \), \( f(x) \to -\infty \)
Both ends go DOWN ↓↓
3. Odd Degree + Positive Leading Coefficient:
As \( x \to -\infty \), \( f(x) \to -\infty \)
As \( x \to +\infty \), \( f(x) \to +\infty \)
Left DOWN, Right UP ↓↑
4. Odd Degree + Negative Leading Coefficient:
As \( x \to -\infty \), \( f(x) \to +\infty \)
As \( x \to +\infty \), \( f(x) \to -\infty \)
Left UP, Right DOWN ↑↓
Zeros and Multiplicity
Multiplicity Rules:
The multiplicity of a zero is the number of times its factor appears in the polynomial.
Graph Behavior at Zeros:
- Odd Multiplicity (1, 3, 5, ...): Graph crosses the x-axis
- Even Multiplicity (2, 4, 6, ...): Graph touches the x-axis and turns around
- Higher Multiplicity: Flatter the graph at the zero
📝 Example:
Analyze zeros and multiplicity: \( P(x) = x^2(x-3)^3(x+2) \)
- \( x = 0 \) with multiplicity 2 (even) → touches x-axis
- \( x = 3 \) with multiplicity 3 (odd) → crosses x-axis
- \( x = -2 \) with multiplicity 1 (odd) → crosses x-axis
Domain and Range of Polynomials
General Rules:
Domain:
All polynomial functions have domain: \( (-\infty, +\infty) \)
Polynomials are defined for all real numbers
Range:
- Odd Degree: Range is \( (-\infty, +\infty) \)
- Even Degree with Positive Leading Coefficient: Range is \( [minimum, +\infty) \)
- Even Degree with Negative Leading Coefficient: Range is \( (-\infty, maximum] \)
Even and Odd Functions
Definitions:
Even Function:
\( f(-x) = f(x) \) for all \(x\)
- Symmetric about the y-axis
- Examples: \( f(x) = x^2 \), \( f(x) = x^4 \), \( f(x) = \cos(x) \)
Odd Function:
\( f(-x) = -f(x) \) for all \(x\)
- Symmetric about the origin (180° rotational symmetry)
- Examples: \( f(x) = x^3 \), \( f(x) = x^5 \), \( f(x) = \sin(x) \)
How to Test:
- Step 1: Find \( f(-x) \) by replacing \(x\) with \(-x\)
- Step 2: Simplify \( f(-x) \)
- Step 3: Compare with \( f(x) \):
- If \( f(-x) = f(x) \) → Even function
- If \( f(-x) = -f(x) \) → Odd function
- Otherwise → Neither even nor odd
📝 Example:
Determine if \( f(x) = 2x^4 - 3x^2 + 5 \) is even, odd, or neither.
Step 1: Find \( f(-x) \)
\( f(-x) = 2(-x)^4 - 3(-x)^2 + 5 = 2x^4 - 3x^2 + 5 \)
Step 2: Compare
\( f(-x) = f(x) \)
Answer: \( f(x) \) is an EVEN function
⚡ Quick Summary
- Use factoring and zero product property to solve polynomial equations
- Roots from factored form: set each factor equal to zero
- Rational Root Theorem: possible rational roots = \( \pm \frac{p}{q} \)
- Descartes' Rule: Count sign changes for positive/negative roots
- Complex roots come in conjugate pairs (with real coefficients)
- Degree \(n\) polynomial has exactly \(n\) roots (Fundamental Theorem)
- End behavior depends on degree (even/odd) and leading coefficient (+/−)
- Multiplicity: odd = crosses x-axis, even = touches x-axis
- Domain of all polynomials is \( (-\infty, +\infty) \)
- Even function: \( f(-x) = f(x) \), symmetric about y-axis
- Odd function: \( f(-x) = -f(x) \), symmetric about origin
📚 Important Formulas Reference
Quadratic Formula:
\( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \)
Sum and Product of Roots (for \( ax^2+bx+c=0 \)):
Sum: \( r_1 + r_2 = -\frac{b}{a} \)
Product: \( r_1 \cdot r_2 = \frac{c}{a} \)
Factored Form from Roots:
\( P(x) = a(x-r_1)(x-r_2)\cdots(x-r_n) \)
Complex Conjugate:
If \( z = a+bi \), then \( \bar{z} = a-bi \)