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When Can You Factor a Quadratic Form?

Brian G. Kronenthal and Felix Lazebnik
Mathematics Magazine
Vol. 87, No. 1 (February 2014), pp. 25-36
DOI: 10.4169/math.mag.87.1.25
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Page Count: 12
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When Can You Factor a Quadratic Form?
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Summary Consider the problem of determining, without using a computer or calculator, whether a given quadratic form factors into the product of two linear forms. A solution derived by inspection is often highly nontrivial; however, we can take advantage of equivalent conditions. In this article, we prove the equivalence of five such conditions. Furthermore, we discuss vocabulary such as “reducible,” “degenerate,” and “singular” that are used in the literature to describe these conditions, highlighting the inconsistency with which this vocabulary is applied.

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