Factoring Polynomials

1) Standard form (arrange polymonial in order of decreasing powers) 

  Example:  ax2+bx+c

2) GCF (Greatest Common Factor or common monomial factor)
What divides into all of the terms exactly (divisible)?

  Examples:  x2+2x     3x2-12x      50x2-48     72x4-12x3+18x2    axn+abxn-1
             x(x+2)    3x(x-4)    2(25x2-24)    6x2(12x2-2x+3)    axn-1(x+b)

3) Difference of two squares (binomial - this step may repeat)
Take the square root of the a and c terms. One factor is the sum of the square roots, the other is the difference.

  Example:    x2-9         25x2-16          16x4-625           a2x2-b2y2
           (x+3)(x-3)   (5x+4)(5x-4)    (4x2+25)(4x2-25)    (ax+by)(ax-by)
                                      (4x2+25)(2x+5)(2x-5)

4) Perfect square (trinomial - may repeat back to step 3) Take the square root of the a and c terms. The b term is twice the product of the terms of the two identical factors.

  Examples:   x2+2x+1    9x2-24x+16     256x4-800xy+625y4       ax2+abxy+b2
            (x+1)(x+1)  (3x-4)(3x-4)  (16x2-25y2)(16x2-25y2)  (ax+by)(ax+by)
             or (x+1)2   or (3x-4)2      or (16x2-25y2)2        or (ax+by)2
Hints:
a) The sign of the linear (first-power) term controls the sign of the second term in the factors.
b) It is easier to assume that the given trinomial is a perfect square and to check after factoring whether the middle term of the trinomial is twice the product of the square roots used in the factors than to use trinomial factoring to get the factors in the first place.

5a) Factorable trinomial; a=1 (may repeat back to step 3) What factors of c add up to b?

  Examples:   x2+3x+2     x2-5x+6     x2+2x-15    x2-x-12    x2+(a-b)x-ab
            (x+2)(x+1)  (x-3)(x-2)  (x+5)(x-3)  (x+3)(x-4)   (x+a)(x-b)

5b) Factorable trinomial; a>1 (may repeat back to step 3) What combination of factors of a and factors of c add up to b?

  Examples:   3x2+7x+2      x2-16x+15      48x2-6x-15    abx2+(bc-ad)x-cd
            (3x+1)(x+2)   (2x-3)(2x-5)   (5x+4)(3x-2)     (ax+c)(bx-d)

Last updated 4/8/97 - 5/31/2000.

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