# Zero-product Property - Application To Finding Roots of Polynomials

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 ).

### Famous quotes containing the words application to, roots, application and/or finding:

The receipt to make a speaker, and an applauded one too, is short and easy.—Take of common sense quantum sufficit, add a little application to the rules and orders of the House, throw obvious thoughts in a new light, and make up the whole with a large quantity of purity, correctness, and elegancy of style.
Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)

There is nothing but is related to us, nothing that does not interest us,—kingdom, college, tree, horse, or iron show,—the roots of all things are in man.
Ralph Waldo Emerson (1803–1882)

My business is stanching blood and feeding fainting men; my post the open field between the bullet and the hospital. I sometimes discuss the application of a compress or a wisp of hay under a broken limb, but not the bearing and merits of a political movement. I make gruel—not speeches; I write letters home for wounded soldiers, not political addresses.
Clara Barton (1821–1912)

It must
Be the finding of a satisfaction, and may
Be of a man skating, a woman dancing, a woman
Combing. The poem of the act of the mind.
Wallace Stevens (1879–1955)