Math CentralQuandaries & Queries


Question from Tori, a student:

How do you factor these types of questions:

* x^3 - 4x^2 - x + 4
*2x^3 - 3x^2 - 4x - 6
*x^4 - 15x^2 - 16

There is a "cubic formula" (in fact two versions, a radical version and one using trig functions) and a "quartic formula" but they are usually more trouble than less powerful methods.

If there are repeating patterns in a polynomial they may indicate a factorization in the same way that (say) 424242 = 10101 * 42.
This method will work on the first problem - you can see a factor of x-4.

Once you find one factor, divide it out and the quotient will be lower-order.

IF the coefficients of the factors are as (Ax+a)(Bx+b)(Cx+c)
then the cubic term is ABCx3 and the constant term is abc. If we suppose that A,B,C,a,b,c are integers, this gets us down to a small number of possibilities; we then check the quadratic term


and the linear term


to see which combination works. That should get the second one (not without some effort!)

Finally, a quartic with no odd (x or x3) terms can often be solved by setting y=x2 to get a quadratic in y. The roots of this may be of the form Ax2 + a and irreducible; or they may be of the form Ax2 - a and factor further. This will work for your example.

Over the reals not every quadratic factors; you can use other tricks such as

x4 + x2 + 1 = (x4 + 2x2 +1) - x2
= [(x2 + 1) + x][x2 +1 -x]
= [x2+x+1][x2-x+1]

Good Hunting!

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