LESSON 22
POLYNOMIALS
Working with Multi-Term Expressions
What is a Polynomial?
A polynomial is an algebraic expression made up of one or more terms, where each term is a number, a variable, or a product of numbers and variables with whole-number exponents. Examples include 3x², 2x + 5, and 4x³ − 7x² + 2x − 1. The word 'polynomial' comes from the Greek 'poly' (many) and Latin 'nomen' (name or term).
Polynomials are the building blocks of algebra. They are used to model curves, describe physical motion, calculate areas of irregular shapes, and form the basis of calculus. Learning to add, subtract, multiply, and factor polynomials gives you powerful tools for solving complex equations and understanding how quantities change.
Naming and Classifying Polynomials
Polynomials are classified by the number of terms: a monomial has one term (5x²), a binomial has two terms (3x + 4), and a trinomial has three terms (x² + 2x − 1). The degree of a polynomial is the highest exponent of the variable. A constant has degree 0, a linear polynomial has degree 1, a quadratic has degree 2, and a cubic has degree 3. Standard form lists terms from highest to lowest degree.
Adding and Subtracting Polynomials
To add or subtract polynomials, combine like terms — terms that have the same variable raised to the same power. Line up like terms vertically or group them horizontally, then add or subtract their coefficients. When subtracting, distribute the negative sign to every term in the second polynomial before combining. The variables and exponents never change — only the coefficients are added or subtracted.
Multiplying Polynomials
To multiply a monomial by a polynomial, use the distributive property — multiply the monomial by each term. To multiply two binomials, use FOIL: First, Outer, Inner, Last — multiply each term in the first binomial by each term in the second, then combine like terms. For larger polynomials, distribute each term of the first polynomial across every term of the second.
Factoring Polynomials
Factoring is the reverse of multiplying — it means rewriting a polynomial as a product of simpler expressions. Always look for a Greatest Common Factor (GCF) first. For trinomials of the form x² + bx + c, find two numbers that multiply to c and add to b. For a difference of squares, use a² − b² = (a + b)(a − b). Factoring is essential for solving quadratic equations and simplifying rational expressions.
Key Facts
The quadratic formula — used to solve any degree-2 polynomial — was known to Babylonian mathematicians around 2000 BCE. They solved it geometrically by 'completing the square,' the same technique still taught today.
Every polynomial of degree n has exactly n roots (solutions), counting repeated roots and complex numbers. This is the Fundamental Theorem of Algebra, proved by Carl Friedrich Gauss in 1799 at age 21.
Polynomials are used in computer graphics to draw smooth curves. Bézier curves — the curves behind every font, logo, and animation on your screen — are defined by polynomial equations.
The FOIL method only works for two binomials. For larger polynomials, you must distribute every term — FOIL is a special case of the full distributive property, not a universal rule.