How Can You Quickly Determine the Number of Roots a Polynomial
The Fundamental Theorem of Algebra
Theorem: A polynomial of degree n tin have at almost northward distinct real roots.
The usefuleness of the Fundamental Theorem comes from the limits that it sets. At most tells us to stop looking whenever we have found n roots of a polynomial of caste n . There are no more.
For example, we may find – past trial and fault, looking at the graph, or other means – that the polynomial
Since the polynomial has caste 3, we would exist wasting our time looking for others.
Having at virtually northward roots, of course, is no guarantee that a polynomial will actually cross the x-axis this maximum number of allowable times. For instance, the cubic polynomial
The possible number of roots between 0 and due north depends on how we count.
If we count distinct roots (every bit nosotros ordinarily do), and then:
A polynomial of fifty-fifty degree can take any number from 0 to n distinct existent roots.
A polynomial of odd degree tin can have any number from ane to n distinct real roots.
This is of piffling aid, except to tell us that polynomials of odd degree must have at least one real root.
If we count roots according to their multiplicity (see The Factor Theorem), then:
A polynomial of degree n can have merely an even number fewer than due north existent roots.
Thus, when nosotros count multiplicity, a cubic polynomial tin can have only three roots or one root; a quadratic polynomial can take only two roots or zero roots. This is useful to know when factoring a polynomial.
The Fundamental Theorem, in its nigh general form (involving complex numbers), has a long history. Finding the roots of polynomials is an activity that has engaged mathematicians for many centuries.
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Source: http://wmueller.com/precalculus/families/fundamental.html
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