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Factoring Special Products
Some binomials and trinomials are produced by special products — specifically, by squaring binomials or by multiplying conjugates. Because factoring reverses multiplication, recognizing that a polynomial matches the output of one of these special product patterns makes it possible to apply the pattern in reverse and factor the polynomial much more quickly than by trial and error.
Two special product patterns from multiplication are especially useful for factoring:
- Squaring a binomial produces a trinomial of the form or (a perfect square trinomial). Recognizing this structure allows the trinomial to be factored back into or .
- Multiplying conjugates produces a binomial of the form (a difference of squares). Recognizing this structure allows the binomial to be factored back into .
The general factoring strategy developed for most polynomials — checking for a GCF, classifying by number of terms, and selecting the appropriate method — still applies. The special products patterns provide additional, faster tools for the specific types of polynomials that arise from these operations.
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