Integer roots
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.
Example:
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(-1)4-2(-1)2-3(-1)-2=0,
(-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=0
has 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(r4-2r2-3r-2)/(r+1)=r3-r2-r-2,
and(r3-r2-r-2)/(r-2)=r2+r+1.
Therefore, the remaining roots must solver2+r+1=0.
The quadratic formula gives In summary, we have found that the roots ofr4-2r2-3r-2=0
are -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, thenP(c)=P'(c)=...=P(k-1)(c)=0,
but 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.r3=-1.
We put -1 into polar form For the third roots, we findr3+1=(r+1)(r2-r+1).
The quadratic formula gives for the roots of the second factor.(r+1)3=0
andr3+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.