Polynomials Cheat Sheet

Polynomial Operations

Addition and Subtraction of Polynomials

  • Objective: Learn vocabulary, identify the degree of a polynomial, and add/subtract polynomials for real-life problem modeling.

Definition of a Polynomial

  • Algebraic expression: Combination of numbers and variables (e.g., x).
  • Monomial: One term (e.g., 3x^4).
  • Polynomial: Several terms (e.g., -7x^5 - 3x^4 + 2x^2 - 5).

Degree of a Polynomial

  • The maximum power of the variable in any term (must be a non-negative integer).
  • Example: Degree of -7x^5 - 3x^4 + 2x^2 - 5 is 5.

Standard Form of a Polynomial

  • Written in decreasing order of power (e.g., ax^5 + bx^4 + … + constant).
  • Constant Term: The term without a variable.
  • Example: 9x^4 - (1/2)x^3 - \pi is a polynomial.

Identifying Polynomials

  • Polynomials have non-negative integer powers.
  • Not Polynomials: Expressions with negative or fractional exponents.
  • Example: x^{-2} + \sqrt{3x} + 1 is not a polynomial.

Leading Term and Coefficient

  • Leading Term: The term with the highest power when in standard form.
  • Leading Coefficient: The number multiplying the variable of the leading term.
  • Example: For -2x^7 + 5x^2 - 2x + 4, the leading coefficient is -2.

Constant Polynomial

  • A number (e.g., 4, 10).
  • Degree is zero (e.g., 10 = 10x^0).

Adding and Subtracting Polynomials

Horizontal Format

  • Combine like terms after removing parentheses.
  • Example: (3x^2 + 2x + 4) + (-x^2 + 2x - 4) = 2x^2 + 4x

Column Method

  • Align like terms in columns and add.
  • Watch out for missing terms/powers and align accordingly.

Subtraction

  • Distribute the negative sign before combining like terms.
  • Example: (3x^3 - 5x^2 + 3) - (x^3 + 2x^2 - x - 4) = 2x^3 - 7x^2 + x + 7

Multiplication of Polynomials

Objectives

  • Use the distributive property and FOIL method.
  • Learn special product formulas.

Terminology

  • Monomial: One term.
  • Binomial: Two terms.
  • Trinomial: Three terms.

Distributive Property

  • Multiply each term inside the parenthesis by the term outside.
  • Example: 3x (2x - 7) = 6x^2 - 21x
  • Example: 4x^2 (3x - 2x^3 + 1) = 12x^3 - 8x^5 + 4x^2

Box Method

  • Multiply polynomials using a grid.
  • Combine like terms in diagonal.

FOIL Method

  • For multiplying two binomials: First, Outer, Inner, Last.
  • Example: (ax + b)(cx + d) = acx^2 + (ad + bc)x + bd

Special Product Formulas

  • Difference of Squares: (a + b)(a - b) = a^2 - b^2
  • Squaring a Binomial: (a + b)^2 = a^2 + 2ab + b^2, (a - b)^2 = a^2 - 2ab + b^2
  • Cubing a Binomial: Formulas exist for (a + b)^3 and (a - b)^3
  • Product of Sum/Difference of Cubes: Formulas exist.