Factor: x^3-7x-6= 0

Question

Hello! I am stuck with this question.

The only idea I could come up with is to take out the x -> x(x^2-7)-6=0
Any help is appreciated!

Answer 1

Unfortunately there is no straightforward plug-and-chug rule for factoring a cubic polynomial. The one thing that we can say for any polynomial is that if x=r is a root to the polynomial expression, one of its factors will be (x-r). When we deal with a cubic equation, we expect three roots--call them r₁, r₂, and r₃. 

 

It might seem like an arbitrary thing to do (because it is), but one of the quickest ways of finding the roots is to just guess-and-check simple values. I'll try 0, 1, 2, and 3:

 

f(x)= x³-7x-6

f(0)= 0³-7*0-6 = 0-0-6 = -6      x=0 is not a root, so (x-0)=x is not a factor.

f(1)= 1³-7*1-6 = 1-7-6 = -12    x=1 is not a root, so (x-1) is not a factor.

f(2)= 2³-7*2-6 = 8-14-6 = -12   x=2 is not a root, so (x-2) is not a factor.

f(3)= 3³-7*3-6 = 27-21-6 = 0   x=3 is a root, so (x-3) is a factor.

 

We know x³-7x-6 factors into something that looks like (x-3)(........). From here, we have two options:

1. Do polynomial division to figure out what's left over: (.......)=(x³-7x-6)/(x-3). Polynomial division is messy though, and if we can avoid it that would be nice.
2. Keep guessing for more roots!

We found some roots corresponding to positive x, so let's look the other way:

 

f(-1)= (-1)³-7*(-1)-6 = -1+7-6 = 0     x=-1 is a root. (x+1) is a factor.

f(-2)= (-2)³-7*(-2)-6 = -8+14-6 = 0   x=-2 is a root. (x+2) is a factor.

 

We got lucky, and guess and check found the roots fairly quickly. We conclude that x³-7x-6=(x-3)(x+2)(x+1). 

 

Just for the sake of it, multiply the factored polynomial together again just to make sure we've got an acceptable answer.

 

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

 

(x²+2x-3x-6)(x+1)

 

(x²-x-6)(x+1)

 

(x²-x-6)x + (x²-x-6)1

 

x³-x²-6x+x²-x-6

 

x³-7x-6

Answer 2

x³ - 7x - 6 = 0

 

 

According to the rational root theorem, the only possible roots of this equation are ±1 , ±2 , ±3 , and ±6.  Using long division or synthetic division to find the factors, we get

 

(x + 2)(x² - 2x - 3) = 0

 

(x + 2)(x - 3)(x + 1) = 0