If the coefficients of a polynomial are integers, it is natural to look for roots which are also integers. Any such root must divide the constant term. We can often ``guess" one or more roots by trying all possibilities.
r4-2r2-3r-2=0.If there is an integer root, it must divide -2. This leaves only four possibilities: 1, -1, 2, and -2. By plugging in, we find that
(-2)4-2(-2)2-3(-2)-2=12.Therefore -1 and 2 are roots, but 1 and -2 are not.
r4-2r2-3r-2=0has the roots -1 and 2. This tells you that the polynomial must contain the factors r+1 and r-2. You can use the long division algorithm to find
(r3-r2-r-2)/(r-2)=r2+r+1.Therefore, the remaining roots must solve
r2+r+1=0.The quadratic formula gives
In summary, we have found that the roots of
r4-2r2-3r-2=0are -1, 2, and .
r3-r2-r+1=(r-1)2(r+1).Since the factor r-1 appears twice, we call 1 a double root of the polynomial, while -1 is a simple root. Note that
also has a root at r=1. This happens in general: If a polynomial P(r) as a k-fold root at r=c, then
If roots are counted by multiplicity (i.e. a double root counts twice, a triple root three times etc.), then a polynomial of nth degree has n roots.
rn=w,where w is a given number. w may be complex, but the following procedure is important even if w is real. The solution of the equation requires writing w in polar form
That is, if x and y are the real an imaginary parts of w, we want to find and is such a way that and . In other words, and are polar coordinates of the point (x,y) in the Cartesian plane.
Since is an angle, it is not unique; it is determined only up to a multiple of . That is, in the polar form of w, we could have written
instead of . Although, we have
if n>1. To find all nth roots of w, you must consider the representations
where k takes the integer values , 1, ..., n-1 (if you go further, then k=n will give you the same as k=0). The nth roots of w are then
r3=-1.We put -1 into polar form
For the third roots, we find
r3+1=(r+1)(r2-r+1).The quadratic formula gives
for the roots of the second factor.
r3+1=0.The first equation has a triple root at -1, the second has three different roots: -1 and . In general, the equation rn=w always has n DIFFERENT ROOTS. Be prepared for a whipping with the cat of nine tails if you should ever say that the equation rn=w has an n-fold root.