✦ For everyone, free.

Practical knowledge for real and everyday life

Home

1.48 Special Factoring Definitions

Special Factoring Definitions explores key techniques for simplifying polynomials, including methods like factoring trinomials, differences of squares, and grouping.

Special Factoring Definitions establishes the vocabulary for a set of recurring polynomial patterns that factor according to fixed, memorable templates rather than requiring a general search for factor pairs, covering the pattern formed by two subtracted squares, the pattern formed by a squared binomial, and the two parallel patterns formed by the sum and the difference of two cubes.

Difference of Squares

A difference of squares is a binomial of the form a² − b², which always factors into the product of a sum and a difference of the same two terms: (a + b)(a − b). Recognizing an expression as a difference of squares—two perfect square terms separated by subtraction—allows it to be factored immediately using this fixed pattern, without needing to search for factor pairs as would be required for a general trinomial.

a 2 − b 2 = ( a + b ) ( a − b )

Perfect Square Trinomial

A perfect square trinomial is a trinomial that results from squaring a binomial, taking the form a² + 2ab + b² or a² − 2ab + b², which factors into (a + b)² or (a − b)² respectively. A perfect square trinomial can be recognized by checking whether its first and last terms are perfect squares and whether its middle term equals twice the product of the square roots of those first and last terms.

a 2 + 2 a b + b 2 = (a+b) 2

Sum of Cubes

A sum of cubes is a binomial of the form a³ + b³, which factors into the product of a binomial and a trinomial according to the fixed pattern (a + b)(a² − ab + b²). Unlike a difference of squares, a sum of cubes does not factor into two binomials, but rather into one binomial paired with a trinomial that is generally not further factorable using integer coefficients.

a 3 + b 3 = ( a + b ) ( a 2 − a b + b 2 )

Difference of Cubes

A difference of cubes is a binomial of the form a³ − b³, which factors into the product of a binomial and a trinomial according to the fixed pattern (a − b)(a² + ab + b²). The difference of cubes pattern closely mirrors the sum of cubes pattern, differing only in the placement of the two sign changes—between the binomial's terms and within the trinomial's middle term.

a 3 − b 3 = ( a − b ) ( a 2 + a b + b 2 )

Together, these definitions catalog four recognizable polynomial patterns—difference of squares, perfect square trinomial, sum of cubes, and difference of cubes—each with its own fixed factoring template, allowing expressions matching these specific structures to be factored immediately and reliably rather than through a slower, general search process.

Related content