Application To Finding Roots of Polynomials
Suppose and are univariate polynomials with real coefficients, and is a real number such that . (Actually, we may allow the coefficients and to come from any integral domain.) By the zero-product property, it follows that either or . In other words, the roots of are precisely the roots of together with the roots of .
Thus, one can use factorization to find the roots of a polynomial. For example, the polynomial factorizes as ; hence, its roots are precisely 3, 1, and -2.
In general, suppose is an integral domain and is a monic univariate polynomial of degree with coefficients in . Suppose also that has distinct roots . It follows (but we do not prove here) that factorizes as . By the zero-product property, it follows that are the only roots of : any root of must be a root of for some . In particular, has at most distinct roots.
Read more about this topic: Zero-product Property
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