Mathematics.

expressions

Polynomials

Algebra I30 minDifficulty3 out of 10

You should know: exponents

Overview

A polynomial is an expression built from variables and constants using only addition, subtraction, and non-negative integer exponents — no division by a variable, no variables under a radical, and no negative or fractional exponents on variables. Polynomials are the workhorse expressions of algebra: every quadratic, cubic, and higher-degree equation studied in school algebra is a polynomial equation, and polynomial functions are the smoothest, best-behaved functions in mathematics.

Intuition

Think of a polynomial as a sum of 'building blocks,' each block being a number times a whole-number power of x (like 5x³ or -2x or 7). There's no dividing by x, no square roots of x, and no x in an exponent — just clean, additive combinations of powers. The DEGREE (the highest power present) tells you the overall shape: degree 1 is a line, degree 2 is a parabola, degree 3 has up to two turning points, and so on — one fewer turning point than the degree, at most.

Interactive Graph

Graph a cubic polynomial

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Formal Definition

Definition

A polynomial in one variable x is any expression of the form:

P(x)=anxn+an−1xn−1+⋯+a1x+a0,an≠0P(x) = a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0, \quad a_n \neq 0

n is a non-negative integer (the degree), and the aᵢ are constants (coefficients)

General form

Notation

NotationMeaning
deg⁡(P)=n\deg(P)=nThe degree of a polynomial — its highest exponent
ana_nThe leading coefficient — the coefficient of the highest-degree term

Derivation

Multiplying two binomials using the distributive property (FOIL is just a mnemonic for applying distribution twice):

(x+3)(x+5)=x(x+5)+3(x+5)(x+3)(x+5) = x(x+5) + 3(x+5)

Distribute the first binomial over the second

=x2+5x+3x+15= x^2+5x+3x+15

Distribute each term

=x2+8x+15= x^2+8x+15

Combine like terms (5x + 3x = 8x)

Properties

Closure under addition/subtraction

The sum or difference of two polynomials is a polynomial\text{The sum or difference of two polynomials is a polynomial}

Closure under multiplication

The product of two polynomials is a polynomial, with deg⁡(PQ)=deg⁡(P)+deg⁡(Q)\text{The product of two polynomials is a polynomial, with } \deg(PQ)=\deg(P)+\deg(Q)

Degree of a sum

deg⁡(P+Q)≤max⁡(deg⁡P,deg⁡Q)\deg(P+Q) \le \max(\deg P, \deg Q)

FOIL for binomials

(a+b)(c+d)=ac+ad+bc+bd(a+b)(c+d) = ac+ad+bc+bd

Applications

Polynomial models approximate the behavior of physical systems (e.g. stress-strain curves) over limited ranges of input.

Worked Examples

  1. Combine like terms (same power of x).

    (3x2+x2)+(2x−4x)+(−5+7)(3x^2+x^2) + (2x-4x) + (-5+7)

Answer: 4x² - 2x + 2

Practice Problems

Difficulty 2/10

Simplify: (5x³ - 2x + 1) - (2x³ + 3x - 4).

Difficulty 3/10

Multiply (x + 2)(x - 2)(x + 1).

Common Mistakes

Common Mistake

Combining unlike terms, e.g. adding 3x² and 5x to get 8x² or 8x.

Only terms with the EXACT same variable and exponent (like terms) can be combined. 3x² and 5x are different powers of x and cannot be combined.

Common Mistake

Forgetting to distribute a negative sign across an entire polynomial being subtracted, e.g. (5x-2)-(3x-4) treated as 5x-2-3x-4.

Subtracting a polynomial means distributing a factor of -1 across EVERY term: (5x-2)-(3x-4) = 5x-2-3x+4, not 5x-2-3x-4.

Summary

  • A polynomial is a sum of terms of the form aₖxᵏ, with k a non-negative integer.
  • The degree of a polynomial is its highest exponent; the leading coefficient is attached to that term.
  • Polynomials are closed under addition, subtraction, and multiplication.
  • Multiplying polynomials uses the distributive property repeatedly (FOIL is a special case for two binomials).

References

  1. WebsiteWikipedia — Polynomial