# Is it possible to factor #y=x^4 - 2x^3 - 13x^2 + 14x + 24 #? If so, what are the factors?

##### 1 Answer

Use the rational root theorem to help find the first two factors, then divide and factor the remaining quadratic to find:

#y = x^4-2x^3-13x^2+14x+24#

#=(x+1)(x-2)(x-4)(x+3)#

#### Explanation:

Let

By the rational root theorem, any rational roots of

That means that the only possible rational zeros are:

#+-1# ,#+-2# ,#+-3# ,#+-4# ,#+-6# ,#+-8# ,#+-12# ,#+-24#

Try the first few in turn:

#f(1) = 1-2-13+14+24 = 24#

#f(-1) = 1+2-13-14+24 = 0#

#f(2) = 16-16-52+28+24 = 0#

So

#x^4-2x^3-13x^2+14x+24#

#=(x+1)(x^3-3x^2-10x+24)#

#=(x+1)(x-2)(x^2-x-12)#

To factor the remaining quadratic, find a pair of factors of

#=(x+1)(x-2)(x-4)(x+3)#