Study Notes: Polynomials

Overview

Polynomials form a cornerstone of algebra and appear regularly in SOF IMO, both as direct questions and within multi-step reasoning problems. This topic covers polynomial expressions, their zeroes (roots), and two fundamental theorems—remainder theorem and factor theorem—that help factorize and evaluate polynomials efficiently. Mastering these concepts is essential because they connect to quadratic equations, coordinate geometry, and real-world problem modeling in higher classes.

For SOF IMO, expect 3–5 questions testing your ability to find zeroes, apply the remainder or factor theorem, verify algebraic identities, and manipulate polynomial expressions. The Achievers Section often embeds polynomial problems in word-problem or higher-order thinking contexts. A strong grasp of definitions, standard identities, and theorem applications will give you speed and accuracy.

Key Concepts

• **Polynomial definition**: An algebraic expression of the form p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where aₙ ≠ 0, n is a non-negative integer, and coefficients are real numbers.
• **Degree of a polynomial**: The highest power of the variable with a non-zero coefficient. A constant (non-zero) has degree 0; zero polynomial has no defined degree.
• **Zero (root) of a polynomial**: A value α such that p(α) = 0. A polynomial of degree n has at most n real zeroes.
• **Remainder Theorem**: When polynomial p(x) is divided by (x − a), the remainder is p(a). This lets you find remainders without performing long division.
• **Factor Theorem**: (x − a) is a factor of p(x) if and only if p(a) = 0. This is the converse logic used to factorize polynomials.
• **Relationship between zeroes and coefficients**: For a quadratic ax² + bx + c, if α and β are zeroes, then α + β = −b/a and αβ = c/a. Similar relations exist for cubic polynomials.
• **Algebraic identities**: Pre-memorized expansion formulas like (a + b)² = a² + 2ab + b², (a − b)(a + b) = a² − b², and (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca speed up factorization and simplification.

Formulas / Key Facts

1. **Standard identities**:

• (a + b)² = a² + 2ab + b²
• (a − b)² = a² − 2ab + b²
• a² − b² = (a − b)(a + b)
• (x + a)(x + b) = x² + (a + b)x + ab
• (a + b)³ = a³ + b³ + 3ab(a + b)
• (a − b)³ = a³ − b³ − 3ab(a − b)
• a³ + b³ = (a + b)(a² − ab + b²)
• a³ − b³ = (a − b)(a² + ab + b²)

2. **Quadratic zeroes and coefficients**: For p(x) = ax² + bx + c, sum of zeroes = −b/a, product of zeroes = c/a.

3. **Cubic zeroes and coefficients**: For p(x) = ax³ + bx² + cx + d with zeroes α, β, γ:

• α + β + γ = −b/a