Polynomials - Formulas & Theorems
IB Mathematics Analysis & Approaches (SL & HL)
📐 Polynomial Definition
General Form:
A polynomial of degree \(n\) has the form:
\[P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_2x^2 + a_1x + a_0\]
where \(a_n \neq 0\), \(a_i\) are real constants, and \(n\) is a non-negative integer
Key Terms:
Degree: The highest power of \(x\) (which is \(n\))
Leading Term: The term with the highest power (\(a_nx^n\))
Leading Coefficient: The coefficient of the leading term (\(a_n\))
Common Polynomial Types:
• Degree 1: Linear → \(ax + b\)
• Degree 2: Quadratic → \(ax^2 + bx + c\)
• Degree 3: Cubic → \(ax^3 + bx^2 + cx + d\)
• Degree 4: Quartic → \(ax^4 + bx^3 + cx^2 + dx + e\)
🔑 Factor Theorem
Statement:
For a polynomial \(P(x)\) and a constant \(k\):
\[P(k) = 0 \iff (x - k) \text{ is a factor of } P(x)\]
Usage:
• If \(P(k) = 0\), then \((x - k)\) is a factor
• If \((x - k)\) is a factor, then \(P(k) = 0\)
• Used to find factors and roots of polynomials
📊 Remainder Theorem
Statement:
When a polynomial \(P(x)\) is divided by \((x - k)\), the remainder is \(P(k)\).
\[\text{Remainder} = P(k)\]
General Form:
\[P(x) = (x - k)Q(x) + R\]
where \(Q(x)\) is the quotient and \(R\) is the remainder
Extended Form (for linear divisors):
When dividing by \((ax - b)\):
\[\text{Remainder} = P\left(\frac{b}{a}\right)\]
➗ Polynomial Division
Division Algorithm:
\[P(x) = D(x) \cdot Q(x) + R(x)\]
where \(D(x)\) = divisor, \(Q(x)\) = quotient, \(R(x)\) = remainder
The degree of \(R(x)\) is less than the degree of \(D(x)\)
Methods:
• Long Division: Traditional algorithm for dividing polynomials
• Synthetic Division: Shortcut method for dividing by linear factors
• Factor Theorem: For finding remainders quickly
🎯 Roots, Zeros, and Factors
Definitions:
Zero/Root: A value \(x = r\) where \(P(r) = 0\)
Factor: If \(r\) is a root, then \((x - r)\) is a factor
Multiplicity: The number of times a root is repeated
Number of Roots:
A polynomial of degree \(n\) has exactly \(n\) roots (counting multiplicities)
These roots may be real or complex, and may be repeated
➕ Sum and Product of Roots (HL)
For polynomial: \(a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 = 0\)
With roots \(\alpha_1, \alpha_2, \ldots, \alpha_n\):
Sum of Roots:
\[\sum_{i=1}^{n} \alpha_i = -\frac{a_{n-1}}{a_n}\]
Given in formula booklet as: \(\alpha_1 + \alpha_2 + \cdots + \alpha_n = -\frac{a_{n-1}}{a_n}\)
Product of Roots:
\[\prod_{i=1}^{n} \alpha_i = (-1)^n \frac{a_0}{a_n}\]
Given in formula booklet as: \(\alpha_1 \cdot \alpha_2 \cdot \ldots \cdot \alpha_n = (-1)^n \frac{a_0}{a_n}\)
Quadratic Example (degree 2):
For \(ax^2 + bx + c = 0\) with roots \(\alpha\) and \(\beta\):
\[\alpha + \beta = -\frac{b}{a}\]
\[\alpha \cdot \beta = \frac{c}{a}\]
🔢 Complex Roots Theorem (HL)
Conjugate Root Theorem:
If a polynomial has real coefficients and \(a + bi\) is a complex root (where \(b \neq 0\)), then its complex conjugate \(a - bi\) is also a root.
\[\text{If } (a + bi) \text{ is a root, then } (a - bi) \text{ is also a root}\]
Corresponding Quadratic Factor:
If \(a + bi\) and \(a - bi\) are roots, the corresponding factor is:
\[(x - (a + bi))(x - (a - bi)) = (x - a)^2 + b^2\]
🌟 Fundamental Theorem of Algebra
Statement:
• Every polynomial of degree \(n \geq 1\) has exactly \(n\) complex roots (counting multiplicities)
• Every polynomial can be factored into \(n\) linear factors over the complex numbers
• Every real polynomial can be factored into real linear and/or real irreducible quadratic factors
Factored Form:
\[P(x) = a_n(x - \alpha_1)(x - \alpha_2)\cdots(x - \alpha_n)\]
where \(\alpha_1, \alpha_2, \ldots, \alpha_n\) are the roots (may be complex or repeated)
📈 Graph Properties
End Behavior:
Determined by the leading term \(a_nx^n\):
• Even degree, \(a_n > 0\): Both ends go up
• Even degree, \(a_n < 0\): Both ends go down
• Odd degree, \(a_n > 0\): Left down, right up
• Odd degree, \(a_n < 0\): Left up, right down
Turning Points:
A polynomial of degree \(n\) has at most \(n - 1\) turning points
Multiplicity and Graph Behavior:
• Multiplicity 1: Graph crosses the x-axis
• Multiplicity 2: Graph touches (turning point) the x-axis
• Odd multiplicity ≥ 3: Graph crosses with a point of inflection
• Even multiplicity ≥ 4: Graph touches with a turning point
🔨 Forming Polynomial Equations
From Known Roots:
If roots are \(\alpha_1, \alpha_2, \ldots, \alpha_n\), the polynomial is:
\[P(x) = a(x - \alpha_1)(x - \alpha_2)\cdots(x - \alpha_n)\]
where \(a\) is any non-zero constant (usually chosen so leading coefficient is 1 or a specific value)
For Quadratic with Roots \(\alpha\) and \(\beta\):
\[x^2 - (\alpha + \beta)x + \alpha\beta = 0\]
or: \(x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0\)
💡 Exam Tip: The sum and product of roots formulas are given in the IB formula booklet. Factor and Remainder theorems are essential for solving polynomial questions efficiently. Remember that complex roots always come in conjugate pairs for polynomials with real coefficients. Use your GDC for polynomial division and finding roots when allowed.