[PDF] Math 461 F Spring 2011 Descartes Factor Theorem Drew Armstrong

101352_6FactorTheorem.pdf

Math 461 F Spring 2011

Descartes' Factor Theorem Drew ArmstrongDescartes'La Geometrie(1637) is the oldest work of mathematics that

makes sense to our modern eyes, because it was the rst work to use our modern symbolic notation.La Geometrieis famous for introducing the idea of coordinate geometry | indeed the \Cartesian" plane is named after Descartes | but it also contains an important result in the theory of poly- nomials, called the Factor Theorem. I will give a modern treatment of this result. LetFbe anyeld(if you don't like the word \eld" you can think \number system") and letF[x] be theringof polynomials with coecients inF(if you don't like the word \ring" you can just ignore it). Then we have the following. The Factor Theorem.Letf(x)2F[x] be a polynomial of degreenand suppose thatf() = 0 for some2F(we sayis arootoff(x)). Then we can write f(x) = (x)g(x); whereg(x)2F[x] is a polynomial of degreen1.

Proof.For any positive integerdwe have

x dd= (x)'d where'd=xd1+xd2+xd32+


+xd2+d1. To see this, just expand the right side and observe that all the terms cancel exceptxdd. Now suppose thatf(x) =anxn+an1xn1+


+a1x+a0withan6= 0. Sincef() = 0, we may writef(x) =f(x)f(). On the other hand, we have f(x)f() =an(xnn) +an1(xn1n1) +


+a1(x) =an(x)'n+an1(x)'n1+


+a1(x) = (x)[an'n+an1'n1+


+a2'2+a1] = (x)anxn1+ lower terms:  This result is really at the beginning of algebra, and it eventually leads toGalois theory. Let me present a few important consequences. Corollary.Letf(x)2F[x] have degreen. Thenf(x) hasat mostnroots inF. Proof.We will prove this by induction onn. It is certainly true forn= 1 sinceax+b= 0 has the unique solutionx=b=a. Now letf(x)2F[x] have degreek2. Iff(x) has zero roots, we are done. So suppose thatf() = 0 for some2F. By the factor theorem we can writef(x) = (x)g(x), whereg(x)2F[x] has degreek1. But now anyotherroot off(x) must be a root ofg(x), and by inductiong(x) has at mostn1 roots. Hence f(x) has at mostnroots. We say that a eldFisalgebraically closedif every polynomialf(x)2F[x] of degreenhasexactlynrootsinF. Note that the real numbersRarenot algebraically closed because the polynomialx2+ 1 has no real roots. It is a celebrated fact that that the complex numbersCarealgebraically closed, which is called the Fundamental Theorem of Algebra. (I hope to present a proof in this course.) Here is another important corollary of the Factor Theorem. Corollary.Suppose thatf(x) =ax2+bx+chas rootsrands. Then ax

2+bx+c=a(xr)(xs):

As a consequence, we getr+s=b=aandrs=c=a.

Proof.Sincef(r) = 0, the Factor Theorem says thatf(x) = (xr)g(x), whereg(x) is a linear (degree 1) polynomial. Now we must haveg(s) = 0 and the Factor Theorem implies thatg(x) = (xs)h(x), whereh(x) is a degree 0 polynomial. That is,h(x) is just a number, sayh(x) =C2F. We conclude that f(x) =C(xr)(xs) =Cx2C(r+s)x+Crs: But we already know that the coecient ofx2isa, henceC=a. That is, the polynomial (xr)(xs) is theuniquepolynomial with rootsrands. (We could multiply it by a constant, but in the theory of polynomials we don't really care about constant multiples.) Using the same method of proof, we could show the following. Corollary.Letf(x)2F[x] be a polynomial of degreenwhich has a full set of rootsr1;r2;:::;rn2F. It follows that f(x) =C(xr1)(xr2)


(xrn); for some constantC2F. [Note: The results on this handout are extremely important, and you should not pass the course if you do not understand them.]