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:
“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)
“May my application so close
To so endless a repetition
Not make me tired and morose
And resentful of man’s condition.”
—Robert Frost (1874–1963)
“To people off alone, as we were, there is something stirring about finding evidences of human labour and care in the soil of an empty country. It comes to you as a sort of message, makes you feel differently about the ground you walk over every day.”
—Willa Cather (1873–1947)
“Though of erect nature, man is far above the plants. For man’s superior part, his head, is turned toward the superior part of the world, and his inferior part is turned toward the inferior world; and therefore he is perfectly disposed as to the general situation of his body. Plants have the superior part turned towards the lower world, since their roots correspond to the mouth, and their inferior parts towards the upper world.”
—Thomas Aquinas (c. 1225–1274)