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College and Advanced Algebra –1st Cheat Sheet (DRAFT) by

1SEM_AY2023–2024_MATH2023

This is a draft cheat sheet. It is a work in progress and is not finished yet.

Order of Operations

BODMAS / PEMDAS
B - Brackets: Perform operations inside parent­heses or brackets first.
O - Orders: Evaluate expres­sions with exponents (powers and roots).
D - Division: Perform division from left to right.
M - Multip­lic­ation: Perform multip­lic­ation from left to right.
A - Addition: Perform addition from left to right.
S - Subtra­ction: Perform subtra­ction from left to right.

SPECIAL PRODUCTS INVOLVING CUBES

The products are just the result of multip­lying out the bracket.
(x + y)3 = x3 + 3x2y + 3xy2 + y3 (Cube of a sum)
(x − y)3 = x3 − 3x2y + 3xy2 − y3 (Cube of a differ­ence)
(x + y)(x2 − xy + y2) = x3 + y3 (Sum of 2 cubes)
(x − y)(x2 + xy + y2) = x3 − y3 (Diffe­rence of 2 cubes
  

Law of Inclusion and Exclusion

PROBLEM SETS
For three sets A, B, and C:
|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|
and
|A ∩ B ∩ C| = |A ∩ B ∩ C| - |A| - |B| - |C| + |A ∩ B| + |A ∩ C| + |B ∩ C|

THE FOIL METHOD

FOIL – is sometimes used to find the product of two binomials.
We multiply,
• the First terms,
• the Outer terms,
• the Inner terms, and then
• the Last terms of each binomials.
  

Venn Diagram

RULES OF EXPONENT

Special Products

Product of a binomial and a trinomial of the form.
(a + b)(c + d + e) = ac + ad + ae + bc + bd + be

Monomial= 2y², 2
Binomial = (2y²-2)
Trinomial = ( 2y³+y²+2)
  

Special Products: Square of a Binomial

•Products in algebra have patterns that occur frequently
• These patterns are called special products

( x + y )² = x² + 2xy + y² (Square of a sum)
( x − y )² = x² − 2xy + y² (Square of a differ­ence)

Special Product

Product of Sum and Difference of Two Terms
(a + b)(a - b) = a2 - b2