# How do you use the rational root theorem to find the roots of #3x^4+2x^3-9x^2-12x-4=0#?

##### 1 Answer

Use the rational root theorem to identify the possible roots, try some and narrow down the search to find roots

#### Explanation:

#f(x) = 3x^4+2x^3-9x^2-12x-4#

By the rational root theorem any rational roots of

So the possible rational roots are:

#+-1/3# ,#+-2/3# ,#+-1# ,#+-4/3# ,#+-2# ,#+-4#

Try:

#f(1/3) = 1/27+2/27-1-4-4 = 1/9-9 = -80/9#

#f(-1/3) = 1/27-2/27-1+4-4 = -28/27#

#f(2/3) = 16/27+16/27-4-8-4 = 32/27#

#f(-2/3) = 16/27 - 16/27-4+8-4 = 0#

So

#+-1# ,#+-2#

Try:

#f(1) = 3+2-9-12-4 = -20#

#f(-1) = 3-2-9+12-4 = 0#

#f(2) = 48+16-36-24-4 = 0#

#f(-2) = 48-16-36+24-4 = 16#

So we have identified

To check, we can multiply out the corresponding factors:

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

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

#=3x^4+(5-3)x^3-(6+5-2)x^2-(10+2)x-4#

#=3x^4+2x^3-9x^2-12x-4#