x^3-2x^2-x+2=0 , how can I find x?

Question

I'm really stuck with this on my assignment. I found out that -1, 1 and 2 work just fine. But I need to prove that. Can anyone help?

Answer 1

Hi again Erlandas;

x³-2x²-x+2=0

Let's factor this equation.

For the FOIL...

FIRST must be (x²)(x)=x³

OUTER and INNER must add-up to -2x²-x

LAST...(2)(1) or (1)(2), and both numbers must be either positive or negative.

(x²-1)(x-2)=0

Let's FOIL...

FIRST...(x²)(x)=x³

OUTER...(x²)(-2)=-2x²

INNER...(-1)(x)=-1x=-x

LAST...(-1)(-2)=2

x³-2x²-x+2=0

So...

(x²-1)(x-2)=0

Either of these parenthetical equations must =0...

x²-1=0

x²=1

x=1 or -1

or...

x-2=0

x=2

 

Let's check our results with the original equation...

x³-2x²-x+2=0

x=1

1³-[2(1²)]-1+2=0

1-[(2)(1)]-1+2

1-2-1+2

0=0

 

x=2

x³-2x²-x+2=0

2³-[2(2²)]-2+2

8-[(2)(4)]-2+2

8-8-2+2=0

0=0