   SEARCH HOME Math Central Quandaries & Queries  Question from peter, a student: hello I,am having trouble factorising a polynomial into polynomial factors with real coefficients please can you help the polynomial is x^2+729 Thanks Hi Peter,

Suppose the problem was to factor x2 - 729. This I can do since 729 = 272 and thus x2 - 729 = x2 - 272 is a difference of squares and thus

x2 - 729 = x2 - 272 = (x - 27)(x + 27)

In particular this means that if x is an integer then the integer x2 - 729 factors. For example if x = 2 then

x2 - 729 = 22 - 729 = -725 = -25 × 29.

Now what about x2 + 729? In this case if x = 2 then x2 + 729 = 733 which is a prime and hence doesn't factor except as 1 × 733. Thus the only possible factorization of x2 + 729 is 1 × (x2 + 729) and hence we say that x2 + 729 doesn't factor.

Harley

Peter wrote back

Question from peter, a student:

I,am having trouble factorising a polynomial x^6+729 into polynomial factors,
with real coefficients, where the factors are either linear or quadratic.
you have explained the process when x is x^2 what happens when x is x^3 or x^4
etc.

thanks Pete

Since 729 = 93, if you write z = x2 then

x6 + 729 = z3 + 93

which is a sum of cubes. Sums of cubes do factor nicely

z3 + a3 = (z + a)(z2 - az + a2)

Harley     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.