SOF IMO · Bite-size lessons

Polynomials

About 6 min · Mathematical Reasoning · English

Time worked out from the lesson's 396 words at 150 words a minute, plus a minute for each question.

Polynomials form a cornerstone of algebra and can appear 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.

Key ideas

• 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.

Watch out for

• 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.
• 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.
• 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.

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).

Check yourself: 3 questions

This lesson is cut from the topic's note, not rewritten. Fact-checked on 21 Oct 2026: an AI-assisted check of this note against cited sources; the errors it found were corrected that day. Some errors may remain. The questions are from Shishya's validated question bank.

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