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
• αβ + βγ + γα = c/a
• αβγ = −d/a

4. **Remainder Theorem**: Remainder when p(x) is divided by (x − a) equals p(a).

5. **Factor Theorem**: p(a) = 0 ⟺ (x − a) divides p(x).

6. **Division algorithm for polynomials**: p(x) = g(x)·q(x) + r(x), where degree of r(x) < degree of g(x).

Worked Examples

**Example 1: Find the zeroes of p(x) = 2x² − 5x + 3 and verify the relationship.**

*Solution*: Factorize: 2x² − 5x + 3 = 2x² − 2x − 3x + 3 = 2x(x − 1) − 3(x − 1) = (2x − 3)(x − 1). Zeroes: 2x − 3 = 0 ⇒ x = 3/2; x − 1 = 0 ⇒ x = 1. Sum of zeroes = 3/2 + 1 = 5/2 = −(−5)/2 = −b/a ✓ Product of zeroes = (3/2)(1) = 3/2 = c/a ✓

**Example 2: Use the remainder theorem to find the remainder when p(x) = x³ − 4x² + 6x − 8 is divided by (x − 2).**

*Solution*: By remainder theorem, remainder = p(2). p(2) = (2)³ − 4(2)² + 6(2) − 8 = 8 − 16 + 12 − 8 = −4. Remainder is −4.

**Example 3: Show that (x + 1) is a factor of x³ + 2x² − x − 2.**

*Solution*: By factor theorem, check if p(−1) = 0. p(−1) = (−1)³ + 2(−1)² − (−1) − 2 = −1 + 2 + 1 − 2 = 0. Since p(−1) = 0, (x + 1) is a factor.

**Example 4: Find a quadratic polynomial whose zeroes are 3 and −2.**

*Solution*: Sum of zeroes = 3 + (−2) = 1; product = 3·(−2) = −6. Polynomial: x² − (sum)x + (product) = x² − x − 6.

Common Mistakes

1. **Confusing remainder with quotient** → Remember: remainder theorem gives you p(a) directly, not the quotient. Don't perform long division if only remainder is asked.

2. **Forgetting to simplify before applying identities** → Always group or factor common terms first. Jumping straight to identities on unsimplified expressions wastes time and causes errors.

3. **Sign errors in sum and product formulas** → For ax² + bx + c, sum of zeroes is −b/a (note the negative). Writing b/a is the most common slip.

4. **Assuming a zero means the entire polynomial is zero** → If p(a) = 0, only (x − a) is a factor; you must divide to find the other factors. Don't conclude the polynomial equals zero.

5. **Mixing up factor theorem conditions** → (x − a) is a factor ⟺ p(a) = 0. Students often test p(−a) when the factor is (x − a). Always substitute the opposite sign of the constant in the binomial.

Quick Reference

• **Degree n polynomial has at most n real zeroes**.
• **Remainder when p(x) ÷ (x − a) = p(a)**.
• **(x − a) is a factor ⟺ p(a) = 0**.
• **Quadratic zeroes**: sum = −b/a, product = c/a.
• **Key identity for factorization**: a² − b² = (a − b)(a + b).
• **To construct polynomial from zeroes**: p(x) = x² − (sum)x + (product) for quadratics.