Polynomials: Basic Operations

Definition

A polynomial is an algebraic expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. A monomial is a single term ($ax^k$), and a polynomial is the sum of such terms.

Why It Matters

Polynomials are the “Integers of Algebra.” They are the basic material from which complex functions are built. If you fumble the basic operations (factoring, division), you are stuck at the “Alphabet” level of math, unable to read the “Books” of engineering or economics. Mastery of these operations is the “Baseline Competency” for any technical field.

Core Concepts

• Standard Form: Writing terms in descending order of their degree: $a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$.
• Degree: The highest exponent of the variable in the polynomial.
• Special Product formulas:
  1. Difference of Squares: $(A + B)(A - B) = A^2 - B^2$
  2. Perfect Square Trinomials: $(A \pm B)^2 = A^2 \pm 2AB + B^2$
  3. Sum of Cubes: $(A + B)(A^2 - AB + B^2) = A^3 + B^3$
  4. Difference of Cubes: $(A - B)(A^2 + AB + B^2) = A^3 - B^3$
  5. Cube of a Binomial: $(A \pm B)^3 = A^3 \pm 3A^2B + 3AB^2 \pm B^3$
  • How to read: “The product of the sum and difference of A and B is equal to A squared minus B squared; the square of the binomial A plus or minus B is equal to A squared plus or minus two times A times B plus B squared; the product of A plus B and the quadratic A squared minus A times B plus B squared is equal to A cubed plus B cubed; the product of A minus B and the quadratic A squared plus A times B plus B squared is equal to A cubed minus B cubed; and the cube of the binomial A plus or minus B is equal to A cubed plus or minus three times A squared times B plus or minus three times A times B squared plus or minus B cubed.”
  • Meaning / when to use: Recognize these forms to factor or expand quickly without FOIL—reverse of common products.
• Factoring Strategy:
  1. Factor out the Greatest Common Factor (GCF).
  2. Check the number of terms:
     • 2 terms: Use Difference of Squares, Sum of Cubes, or Difference of Cubes.
     • 3 terms: Check for Perfect Square Trinomial or use $ac$-method/factoring by inspection.
     • 4 terms: Try Factoring by Grouping.
• Division Algorithm: For a dividend $f(x)$ and divisor $g(x)$, there exist unique $q(x)$ (quotient) and $r(x)$ (remainder) such that: $f(x) = g(x)q(x) + r(x)$
  • How to read: “The polynomial f of x is equal to the divisor polynomial g of x times the quotient polynomial q of x, plus the remainder polynomial r of x.”
  • Meaning: Division algorithm—every polynomial divides with a unique quotient and remainder (degree of $r$ less than $g$).

Connected Concepts

• Synthetic Division
• Rational Expressions
• The Binomial Theorem
• Zeros of Polynomial Functions
• First Principles Thinking
• Map-Territory Relation
• Symmetry