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.