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.
If however is not an integral domain, then the conclusion need not hold. For example, the cubic polynomial has six roots in (though it has only three roots in ).
Read more about this topic: Zero-product Property
Famous quotes containing the words application to, application, finding and/or roots:
“If you would be a favourite of your king, address yourself to his weaknesses. An application to his reason will seldom prove very successful.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)
““Five o’clock tea” is a phrase our “rude forefathers,” even of the last generation, would scarcely have understood, so completely is it a thing of to-day; and yet, so rapid is the March of the Mind, it has already risen into a national institution, and rivals, in its universal application to all ranks and ages, and as a specific for “all the ills that flesh is heir to,” the glorious Magna Charta.”
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)
“We are finding out that what looked like a neglected house a year ago is in fact a ruin.”
—Václav Havel (b. 1936)
“April is the cruellest month, breeding
Lilacs out of the dead land, mixing
Memory and desire, stirring
Dull roots with spring rain.”
—T.S. (Thomas Stearns)