Factoring Polynomials and Quadratic Expressions

This set of flashcards covers polynomial factoring techniques including GCF, grouping, trinomials, perfect square identities, and the difference of squares and cubes as presented in the lecture notes.

GCF (Greatest Common Factor)

The initial step in factoring, as seen in the expression $2x^2 + 20x + 48$ where $2$ is pulled out to leave $2(x^2 + 10x + 24)$, which then factors to $2(x+6)(x+4)$.

Factoring by Grouping

A technique used for polynomials with four terms, such as $2x^3 - 6x^2 + 5x - 15$, which is grouped into $2x^2(x-3) + 5(x-3)$ to result in $(x-3)(2x^2 + 5)$.

Trinomial Factoring ($x^2 + 7x + 12$)

A quadratic expression that simplifies into the product of two binomials: $(x+3)(x+4)$.

Coefficient Multiply Method

A process for factoring trinomials where the leading coefficient is multiplied by the constant (e.g., $15 \times -6 = -90$) to split the middle term, as shown in $15x^2 - 9x + 10x - 6 = (3x+2)(5x-3)$.

Perfect Square Trinomial Formula

An algebraic identity where $A^2 + 2AB + B^2 = (A + B)(A + B) = (A + B)^2$, exemplified by $4x^2 + 12x + 9 = (2x + 3)^2$.

Difference of Squares Formula

A factoring rule stated as $A^2 - B^2 = (A + B)(A - B)$, which can be applied to terms like $x^2 - 25 = (x-5)(x+5)$ or $3x^2 - 48 = 3(x-4)(x+4)$.

Factoring Higher Power Difference of Squares

The breakdown of expressions like $16x^4 - 81y^4$ into $(4x^2 - 9y^2)(4x^2 + 9y^2)$ and further into $(2x - 3y)(2x + 3y)(4x^2 + 9y^2)$.

Difference of Cubes Formula

The algebraic rule for cubes defined as $A^3 - B^3 = (A - B)(A^2 + AB + B^2)$, used for examples like $8x^3 - 27$.

Grouping with Perfect Squares

A method used to solve complex expressions like $4x^2 + 20x + 25 - (9y^2 + 24y + 16)$, which converts to $(2x+5)^2 - (3y+4)^2$ and is then solved using the difference of squares: $[(2x+5) + (3y+4)][(2x+5) - (3y+4)]$.