Quandaries
and Queries 

Hello


Hi Marcin, In many instances factoring is recognizing patterns. For example in your first problem I see that 8, a^{3}, 64, b^{3} and c^{6} are all cubes.Hence I can write
Can you factor it now as a sum of cubes? In b), all the coefficients are divisible by 4 and there is an x common factor in each term, thus
Hence you are left with the cubic x^{3} 3x^{2} 11x + 6 which is a candidate for the factor theorem. To use the factor theorem you need to find a root, that is a number x for which
Usually we look for a root that is an integer. Such an integer must divide the constant term (6 in this case). Thus the only candidates for x are 1,2,3,6,1,2,3 and 6. If you try these numbers (start with the small numbers to make the calculation easy) you will find that x=2 is a root. The factor theorem then implies that (x+2) is a factor of the polynomial. Now divide (x+2) into x^{3} 3x^{2} 11x + 6. to get
In this example you can also factor the quadratic. Cheers, 

