1. Find the rational roots of x 4 + 8x 3 + 7x 2 –40x–60 = 0.

Question

1.   Find the rational roots of x 4 + 8x 3 + 7x 2 –40x–60 = 0.

amtran_bus · Answer

I would begin by factoring.

anonymous · Answer

try all numbers that divide $60$
i would try $-1$ first

anonymous · Answer

Okay

anonymous · Answer

unfortunately $-1$ does not work
try $-2$ instead, that does work

anonymous · Answer

{±factors of last term/factors of first term} 

{±1,2,3,4,5,6,10,12,15,20,30,60/1} 

Test each root to see which one works. Try -6 and apply synthetic division. 

1...8...7...-40...-60 
....-6.-12..30....60 
1...2..-5...-10....0 

Rewrite the remaining expression. 

x³ + 2x² - 5x - 10 = 0 

Factor. 

x²(x + 2) - 5(x + 2) = 0 

Combine. 

(x² - 5)(x + 2) = 0 

Set each by zero and solve for x. 

x² - 5 = 0 and x + 2 = 0 

You get: 

x = {±√5, -2} 

However, ±√5 are not rational because it does not express in fraction. Therefore.... 

The rational roots are -6 and -2. 

Hope that helps!

anonymous · Answer

also $-6$ i think
and those are the only rational ones

anonymous · Answer

hope my explanation helps well explanation and answer!