Can you multiply this polynomial?

GammaGrozza
Mon, 06 Jul 2009 20:12:38 +0000

Here’s my proposed solution:
Substitute t^3 with x. So, (x-1)^4(x+1)^4
This equals (x-1)(x+1)(x-1)(x+1)…
This equals (x^2-1)^4.
Substitute y for x^2, so (y-1)(y-1)(y-1)(y-1)
(y^2-2y+1)(y^2-2y+1)
=y^4-2y^3+y^2-2y^3+4y^2-2y+y^2-2y+1
=y^4-4y^3+6y^2-4y+1
=x^8-4x^6+6x^4-4x^2+1 (substituting x^2 for y)
=t^24-4t^18+6t^12-4t^6+1 (subsituting t^3 for x. you could combine the last two steps by figuring that y=x^2=(t^3)^2=t^6, and subsituting t^6 for y)

Is that right? (I checked, it is). There must be an easier (by which I mean faster) way. Although in actuality, this wouldn’t take that long on paper.