I was learning to find the Highest Common Factor of two compound algebraic expressions using the long division method. However, I encountered the following question which puzzled me to a great extent:
Find the HCF of : $x^3$ - $x^2$ - 5x - 3 and $x^3$ - 4$x^2$ - 11x - 6
On examining the two expressions to find there is not simple factor, we divide $x^3$ - 4$x^2$ - 11x - 6 by $x^3$ - $x^2$ - 5x - 3
Omitting the steps of division, we obtain the remainder : -3$x^2$ - 6x - 3
Further inspecting the remainder, we break it down into simple factors: -3($x^2$ + 2x + 1)
Since the two original expressions had no simple factors, we can safely say that the HCF will also not consist of a simple factor. Now, remembering that every remainder contains the HCF, we reject our simple factor i.e -3 and continue on with $x^2$ + 2x + 1
Making the remainder as our next divisor and our previous divisor as our next dividend, proceed with the division: $\dfrac{$x^3$ - $x^2$ - 5x - 3}{$x^2$ + 2x + 1}$
The remainder thus obtained is : -3$x^2$ - 6x - 3
Again we separate simple factors from it to obtain : -3($x^2$ + 2x + 1 ) --------- (1)
Rejecting the simple factor and considering $x^2$ + 2x + 1, we get our new divisor as $x^2$ + 2x + 1 and new dividend (previous divisor) as $x^2$ + 2x + 1
Clearly, they will divide each other and the remainder will be 0. This, thus states that the expression $x^2$ + 2x + 1 is HCF of the two original expressions.
However, now take a look back: We resolved the remainder into the simple factor i.e -3 [as explained in (1)], but we could also have resolved into: 3 (-$x^2$ - 2x - 1 )
Now, lets continue with this as the remainder
We reject the simple factor and consider (-$x^2$ - 2x - 1 ) as our divisor.
So, the nex dividend becomes: $x^2$ + 2x + 1.
Clearly, they will divide each other and the HCF in this case would come out to be : (-$x^2$ - 2x - 1 )
Since two expressions can have only one highest common factors, which one is it here? Is it (-$x^2$ - 2x - 1 ) or ($x^2$ + 2x + 1 )* ?
