lucajava wrote:
For how many values of \(a\) is \(( x - 1)(x^2 - a^2)(x^2 - a - 1)\) divisible by \(x^2 + x - 2\)?
A. 1
B. 2
C. 3
D. 5
E. None
Let's factor the divisor: \(x^2 + x - 2\) into \((x-1)(x+2)\)
Now we see we can cancel the \((x-1)\) terms from both equations, leaving us with:
Dividend: \((x^2 - a^2)(x^2 - a - 1)\), Divisor: \((x+2)\)
The task now is to make \((x+2)\) appear by altering the value of \(a\)
We can factor the dividend further:
\((x - a)(x+a)(x^2 - a - 1)\)
Now, let's see if we can tackle each part:
If \(a = 2, (x+a) =\)
(x+2)If \(a = -2, (x-a) =\)
(x+2)If \(a = 3, (x^2 - a - 1) = (x^2 - 3 -1) = (x^2-4)\) factored
(x+2)(x-2)
The three possible solutions: 2,-2,3