Logarithm Rules

Basic Binomial Identities

General Binomial Expansions

Multinomial Expansions

Difference of Squares and Higher Powers

Sums and Differences of Powers

Special Identities

Factoring Identities

Basic Algebraic Properties

Basic Binomial Identities

	law	formula	explanation
	Square of Sum	$(a + b)^2 = a^2 + 2ab + b^2$	Square of a binomial sum
	Square of Difference	$(a - b)^2 = a^2 - 2ab + b^2$	Square of a binomial difference
	Cube of Sum	$(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$	Cube of a binomial sum
	Cube of Difference	$(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$	Cube of a binomial difference
	Fourth Power of Sum	$(a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4$	Fourth power of a binomial sum
	Fifth Power of Sum	$(a + b)^5 = a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5$	Fifth power of a binomial sum
	Fifth Power of Difference	$(a - b)^5 = a^5 - 5a^4b + 10a^3b^2 - 10a^2b^3 + 5ab^4 - b^5$	Fifth power of a binomial difference

General Binomial Expansions

	law	formula	explanation
	General Binomial Theorem	$(a + b)^n = \sum_{k=0}^{n} C(n,k) a^{n-k} b^k$	General form where C(n,k) = n!/(k!(n-k)!)

Multinomial Expansions

	law	formula	explanation
	Trinomial Square	$(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ac$	Square of a trinomial
	Trinomial Cube	$(a + b + c)^3 = a^3 + b^3 + c^3 + 3(a^2b + ab^2 + a^2c + ac^2 + b^2c + bc^2) + 6abc$	Cube of a trinomial

Difference of Squares and Higher Powers

	law	formula	explanation
	Difference of Squares	$a^2 - b^2 = (a + b)(a - b)$	Factorization of difference of squares
	Difference of Fourth Powers	$a^4 - b^4 = (a^2 + b^2)(a^2 - b^2) = (a^2 + b^2)(a + b)(a - b)$	Factorization of difference of fourth powers

Sums and Differences of Powers

	law	formula	explanation
	Sum of Cubes	$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$	Factorization of sum of cubes
	Difference of Cubes	$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$	Factorization of difference of cubes
	Sum of Fourth Powers	$a^4 + b^4 = (a^2 + \sqrt{2}ab + b^2)(a^2 - \sqrt{2}ab + b^2)$	Factorization of sum of fourth powers
	Sum of Fifth Powers	$a^5 + b^5 = (a + b)(a^4 - a^3b + a^2b^2 - ab^3 + b^4)$	Factorization of sum of fifth powers
	Difference of Fifth Powers	$a^5 - b^5 = (a - b)(a^4 + a^3b + a^2b^2 + ab^3 + b^4)$	Factorization of difference of fifth powers
	General Difference of Powers	$a^n - b^n = (a - b)(a^{n-1} + a^{n-2}b + ... + ab^{n-2} + b^{n-1})$	General factorization of difference of nth powers
	General Sum of Odd Powers	$a^n + b^n = (a + b)(a^{n-1} - a^{n-2}b + ... - ab^{n-2} + b^{n-1})$ when n is odd	General factorization of sum of odd powers

Special Identities

	law	formula	explanation
	Sophie Germain Identity	$a^4 + 4b^4 = (a^2 + 2ab + 2b^2)(a^2 - 2ab + 2b^2)$	Special factorization for fourth powers
	Sum of Squares with Cross Term	$a^4 + a^2b^2 + b^4 = (a^2 + ab + b^2)(a^2 - ab + b^2)$	Alternative factorization of sum-like fourth power expression
	Symmetric Sum Identity	$a^2 + b^2 + c^2 - ab - bc - ca = \frac{1}{2}[(a-b)^2 + (b-c)^2 + (c-a)^2]$	Symmetric expression in terms of pairwise differences
	Sum of Squares Plus Difference	$(a + b)^2 + (a - b)^2 = 2(a^2 + b^2)$	Sum of squared sum and squared difference
	Difference of Squared Sum and Difference	$(a + b)^2 - (a - b)^2 = 4ab$	Difference between squared sum and squared difference
	Complex Factorization	$a^2 + b^2 = (a + bi)(a - bi)$	Factorization using complex numbers

Factoring Identities

	law	formula	explanation
	Quadratic Factoring	$x^2 + (a + b)x + ab = (x + a)(x + b)$	Standard quadratic factoring form
	General Quadratic Roots	$ax^2 + bx + c = a(x - r_1)(x - r_2)$ where $r_1, r_2$ are roots	General quadratic factorization using roots

Basic Algebraic Properties

	law	formula	explanation
	Distributive Property	$a(b + c) = ab + ac$	Basic distributive property
	Product of Binomials	$(a + b)(c + d) = ac + ad + bc + bd$	Product of two binomials
	Product of Trinomials	$(a + b + c)(d + e + f) = ad + ae + af + bd + be + bf + cd + ce + cf$	Product of two trinomials