# How do you use the rational root theorem to find the roots of #2x^3 + 7x^2 - 77x - 40#?

##### 1 Answer

#### Explanation:

By the rational root theorem, any rational zeros of this polynomial must be expressible in the form

That means that the only possible rational zeros are:

#+-1/2# ,#+-1# ,#+-2# ,#+-5/2# ,#+-4# ,#+-5# ,#+-8# ,#+-10# ,#+-20# ,#+-40#

This is rather a lot of possibilities to try, but trying each in turn we soon find:

#f(-1/2) = -1/4+7/4+77/2-40 = (-1+7+154-160)/4 = 0#

So

#2x^3+7x^2-77x-40 = (2x+1)(x^2+3x-40)#

To factor the remaining quadaratic find a pair of factors of

Hence:

#x^2+3x-40 = (x+8)(x-5)#

So the other two zeros are