Factoring Formulas & Special Products

Difference of Squares Formula

a^2 - b^2 = (a-b)(a+b), applicable where both terms are perfect squares and the sign is subtraction.

Example of Difference of Squares

x^2 - 16 = (x-4)(x+4), demonstrates the Difference of Squares formula.

Difference of Cubes Formula

a^3 - b^3 = (a-b)(a^2+ab+b^2), used for subtracting cubes.

Sum of Cubes Formula

a^3 + b^3 = (a+b)(a^2-ab+b^2), used for adding cubes.

Perfect Square Trinomials

(a+b)^2 = a^2+2ab+b^2 and (a-b)^2 = a^2-2ab+b^2, result from squaring a binomial.

GCF (Greatest Common Factor)

The largest common factor of all terms that can be factored out first before further factoring.

Example of GCF

x^4 - x^2 can be factored as x^2(x^2-1).

Difference of Higher Powers General Idea

a^n - b^n = (a-b)(a^{n-1}+a^{n-2}b+…+b^{n-1}), with the sum being similar but more complex.

Conditions to Apply Factoring

2 terms suggest Difference of Squares or Sum/Difference of Cubes; 3 terms check for Perfect Square Trinomials; 4 terms suggest Factoring by Grouping.

Factoring Example with 3 Terms

x^2 + 6x + 9 = (x+3)^2, illustrating Perfect Square Trinomials.