Factoring Completely

Key Questions

  • Which methods of factoring do you use to factor completely?

    You (try to) divide your number by successive primes #2,3,5,7,11...#

    Example :
    Say you want to factor #756#
    Even number, divide by 2: #756=2*378#
    Even, divide by 2: #756=2*2*189#
    Odd, so try 3: #756=2*2*3*63#
    And again: #756=2*2*3*3*21#
    And again: #756=2*2*3*3*3*7#
    7 is a prime, so you're finished.

    You write the result as #756=2^2*3^3*7#

    Remark :
    You never have to try a prime that is larger than the square root of the number left, because if that would divide evenly, the result is a smaller number, and you would have already tried that.

    MeneerNask · 2 · Jan 5 2015
  • What is Factoring Completely?

    For factoring polynomials, "factoring" (or "factoring completely") is always done using some set of numbers as possible coefficient.

    We say we are factoring "over" the set.

    #x^3 -x^2-5x+5# can be factored
    over the integers as #(x-1)(x^2-5)#

    #x^2-5# cannot be factored using integer coefficients. (It is irreducible over the integers.)

    over the real numbers #x^2-5 = (x-sqrt5)(x+sqrt5)#

    One more:
    #x^2+1# cannot be factored over the real numbers, but over the complex numbers it factors as
    #x^2+1=(x-sqrt(-1))(x+sqr(-1))#

    Also written: #(x-i)(x+i)#

    Jim H · 5 · Apr 18 2015

Questions

Polynomials and Factoring