1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Faster Polynomial Multiplication

  1. Sep 3, 2009 #1
    I can solve:

    (2x+3y-4z)(2x-3y+4z) = [2x +(3y-4z)] [2x -(3y-4z)] = (2x)^2 -(3y-4z)^2 = (2x)^2 - (9y^2 - 24yz - 16z) = 4x^2 - 9y^2 + 24yz + 16z which is fine, but if I try to solve:

    (m^2-m-1)(m^2+m-1) = [m^2 -(m+1)][m^2+(m-1) = (m^2)^ 2 - (m-1)^2 = (m^2)^2 - (m^2 -2m +1) = m^4 - m^2 +2m -1 which is not, and I also tried doing:

    [(m^2 - m)-1] [(m^2+m)-1] = (m^4 - m^2) + 1 = this latest one is near, but anyway it should be m^4-3m^2 + 1

    Doing normal multiplication I get: m^4 +m^3 -m^2 -m^3 -m^2+ m-m^2-m+ 1 = m^4-3m^2 + 1

    So why is it that I have -m^2 when is it really -3m^2?

  2. jcsd
  3. Sep 3, 2009 #2


    User Avatar
    Homework Helper
    Education Advisor
    Gold Member

    Since you have factors which are sums of powers of m, use a lattice system to help keep like terms easily found.
  4. Sep 3, 2009 #3


    User Avatar
    Homework Helper

    You are using the equality: (x + y)(x - y) = x2 - y2, right?

    Notice the bolded part, that's where you went wrong. Since m + 1 does not equal m - 1, so you cannot apply the equality here. Remember that:

    (x + y)(x - y) = x2 - y2

    This is totally wrong, since m2 - m is not the same as m2 + m.

    And you are having 2 minus signs here (the red part). It should be one plus, and one minus instead. Please stick to the formula!!! It's NOT: (x - y)(x - y) = x2 + y2!!!!! This is nowhere near correct!!!

    Instead, what we should use here is:

    (x + y)(x - y) = x2 - y2


    So, back to your problem:

    (m2 - m - 1)(m2 + m - 1)

    Notice the 2 terms -m, and +m, they are of different signs. Let's see if you can find any way to apply the equality (x + y)(x - y) = x2 - y2 here. :)
    Last edited: Sep 3, 2009
  5. Sep 3, 2009 #4


    User Avatar
    Science Advisor

    Because you did it wrong, of course!:wink:
    You have [itex][(m^2- m)-1][(m^2+m)-1][/itex] and did the first part as a "sum and difference product": [itex](m^2- m)(m^2+ m)= (m^2)^2- m^2= m^4- m^2[/itex] which is correct. And, of course, (1)(1)= 1. But you forgot the "middle terms" (the "O" and "I" of "FOIL").
    [tex][(m^2- m)- 1][(m^2+m)- 1]= (m^2- m)(m^2+m)+ (m^2-m)(-1)+ (-1)(m^2+m)+(-1)(-1)[/tex]
    [tex]= (m^4- m^2)- (m^2- m)- (m^2+ m)+ 1= m^4- 3m^2+ 1[/tex]
    (The "m" and "-m" terms cancel.)
  6. Sep 3, 2009 #5
    Alright guys, thanks. :smile: Studying yourself + this forum is a lethal formula for learning about a lot of things.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook