Nernst effect

2013-04-15T13:19:44Z

120.61.50.47:

In [[algebra]], the '''factor theorem''' is a theorem linking factors and [[Zero (complex analysis)|zeros]] of a [[polynomial]]. It is a [[special case]] of the [[polynomial remainder theorem]].<ref>{{citation|first=Michael|last=Sullivan|title=Algebra and Trigonometry|page=381|publisher=Prentice Hall|year=1996|isbn=0-13-370149-2}}.</ref>

The factor theorem states that a polynomial <math>f(x)</math> has a factor <math>(x - k)</math> [[if and only if]] <math>f(k)=0</math> (i.e. <math>k</math> is a root).<ref>{{citation|first1=V K|last1=Sehgal|first2=Sonal|last2=Gupta|title=Longman ICSE Mathematics Class 10|page=119|publisher=Dorling Kindersley (India)|isbn=978-81-317-2816-1}}.</ref>

==Factorization of polynomials==
{{Main|Factorization of polynomials}}
Two problems where the factor theorem is commonly applied are those of factoring a polynomial and finding the roots of a polynomial equation; it is a direct consequence of the theorem that these problems are essentially equivalent.

The factor theorem is also used to remove known zeros from a polynomial while leaving all unknown zeros intact, thus producing a lower degree polynomial whose zeros may be easier to find. Abstractly, the method is as follows:<ref>{{citation|first=R. K.|last=Bansal|title=Comprehensive Mathematics IX|page=142|publisher=Laxmi Publications|isbn=81-7008-629-9}}.</ref>
# "Guess" a zero <math>a</math> of the polynomial <math>f</math>. (In general, this can be ''very hard'', but math textbook problems that involve solving a polynomial equation are often designed so that some roots are easy to discover.)
# Use the factor theorem to conclude that <math>(x-a)</math> is a factor of <math>f(x)</math>.
# Compute the polynomial <math> g(x) = f(x) \big/ (x-a) </math>, for example using [[polynomial long division]] or [[synthetic division]].
# Conclude that any root <math>x \neq a</math> of <math>f(x)=0</math> is a root of <math>g(x)=0</math>. Since the [[polynomial degree]] of <math>g</math> is one less than that of <math>f</math>, it is "simpler" to find the remaining zeros by studying <math>g</math>.

===Example===
Find the factors at
: <math>x^3 + 7x^2 + 8x + 2.</math>

To do this you would use trial and error to find the first x value that causes the expression to equal zero. To find out if <math>(x - 1)</math> is a factor, substitute <math>x = 1</math> into the polynomial above:
: <math>x^3 + 7x^2 + 8x + 2 = (1)^3 + 7(1)^2 + 8(1) + 2</math>
: <math>= 1 + 7 + 8 + 2</math>
: <math>= 18.</math>

As this is equal to 18 and not 0 this means <math>(x - 1)</math> is not a factor of <math>x^3 + 7x^2 + 8x + 2</math>. So, we next try <math>(x + 1)</math> (substituting <math>x = -1</math> into the polynomial):
: <math>(-1)^3 + 7(-1)^2 + 8(-1) + 2.</math>

This is equal to <math>0</math>. Therefore <math>x-(-1)</math>, which is to say <math>x+1</math>, is a factor, and <math>-1</math> is a [[Root of a function|root]] of <math>x^3 + 7x^2 + 8x + 2.</math>

The next two roots can be found by algebraically dividing <math>x^3 + 7x^2 + 8x + 2</math> by <math>(x+1)</math> to get a quadratic, which can be solved directly, by the factor theorem or by the [[quadratic equation]].

: <math>{x^3 + 7x^2 + 8x + 2 \over x + 1} = x^2 + 6x + 2</math>

and therefore <math>(x+1)</math> and <math>x^2 + 6x + 2</math> are the factors of <math>x^3 + 7x^2 + 8x + 2.</math>

==Formal version==
Let <math>f</math> be a one-variable polynomial with coefficients in a commutative ring <math>R</math>, and let <math>a \in R</math>. Then <math>f(a) = 0</math> if and only if <math>f(x)=(x-a)g(x)</math> for some polynomial <math>g</math>. In this case, <math>g</math> is determined uniquely.

As for the problem of algorithmically finding all roots, if <math>f</math> is given and <math>a</math> is known, then <math>g</math> can be computed by [[polynomial long division]]; then one can compute the remaining roots of <math>f</math>, including repeated roots, by factoring <math>g</math>.

==References==
{{reflist}}

{{DEFAULTSORT:Factor Theorem}}
[[Category:Polynomials]]
[[Category:Theorems in algebra]]

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Nernst effect