Example

Factor

f(x) = x⁴ + 11x³ + 41x² +61x + 30

Solution

Descartes rule of signs tells us that any roots have to be negative.  The rational root theorem tells us that any rational root has to be a factor of 30.  Hence the possible rational roots are

        -1, -2, -3, -5, -6, -12, -15, -30

We try -1 first since it is the easiest one.  We use synthetic division:

	_	_	_	_	_	_
-1	|	1	11	41	61	30
			-1	-10	-31	-30
	_	_	_	_	_
		1	10	31	30	0

Since the remainder is zero (the last number), we know that x + 1 is a factor.

Hence

        x⁴ + 11x³ + 41x² +61x + 30  =  (x + 1)(x³ + 10x² + 31x + 30)

Now factor 

        x³ + 10x² + 31x + 30

Again, by the rational root theorem and Descartes, we get the possible rational roots as

        -1, -2, -3, -5, -6, -12, -15, -30

Try -1 again

	_	_	_	_	_
-1	|	1	10	31	30
			-1	-9	-22
	_	_	_	_
		1	9	22	8

Since the remainder is not zero, -1 is not a root of this second polynomial.  Try -2

	_	_	_	_	_
-2	|	1	10	31	30
			-2	-16	-30
	_	_	_	_
		1	8	15	0

Since the remainder is zero, -2 is a root of this second polynomial and we have

        x⁴ + 11x³ + 41x² +61x + 30  =  (x + 1)(x + 2)(x² + 8x + 15)

We could use synthetic division as before to factor the last term, but since it is a quadratic, we can just factor it as we are used to

        x² + 8x + 15  =  (x + 3)(x + 5)

The final result is

        x⁴ + 11x³ + 41x² +61x + 30  =  (x + 1)(x + 2)(x + 3)(x + 5)

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