Algebraic Expressions and Polynomials: Monomials, Identities, Factorization
What you must be able to do
A monomial is a product of numbers and variables with whole-number exponents (e.g. −3x²y). A polynomial is a sum of monomials. You must be able to identify the coefficient, the degree, and like terms; add and subtract polynomials by collecting like terms; multiply monomials (add exponents) and polynomials (distribute every term).
Know the standard identities cold, in both directions: (a + b)² = a² + 2ab + b²; (a − b)² = a² − 2ab + b²; a² − b² = (a − b)(a + b); a³ − b³ = (a − b)(a² + ab + b²).
Factorizing means writing an expression as a product. Always try the common factor first, then look for a difference of squares or a perfect-square trinomial, then grouping.
Traps that appear in questions
- Writing (a + b)² = a² + b² - the middle term 2ab is the most commonly forgotten piece.
- Sign slips in (a − b)², where the middle term is negative.
- Adding exponents when multiplying by a sum instead of distributing: 2x(x + 3) = 2x² + 6x, not 2x² + 3.
- Stopping factorization too early: 2x² − 8 = 2(x² − 4) = 2(x − 2)(x + 2).
- Confusing −x² with (−x)².
Rule of thumb
After factorizing, multiply back mentally to check; after expanding, count terms to be sure nothing was dropped.
Worked example
Simplify 3(x − 2)² − (x + 1)(x − 1). 3(x² − 4x + 4) − (x² − 1) = 3x² − 12x + 12 − x² + 1 = 2x² − 12x + 13.