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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