Factoring Polynomials

Factoring, factoring by grouping, GCF (Greatest Common Factor)
Factoring trinomials and polynomials
Factoring, factoring by grouping, GCF or GCD
Greatest Common Factor (or Greatest Common Divisor or Highest Common Factor)
The greatest common factor (GCF) is the greatest factor that divides two given numbers.
The greatest common factor of a polynomial is the largest monomial that divides each term of the polynomial.
Examples:   a)   2a2x - 4a = 2a · (ax - 2)
b)   -2x3 + 4x2 - 10x = -2x · (x2 - 2x + 5)
c)   3a - 3a2b + 6a2c = 3a2 · (a - b + 2c)
d)   -15a4b6 - 5a5b4 + 10a3b5 = -5a3b4 · (3ab2 + a2 - 2b)
Factoring by grouping
Examples:   a)   x2 - 1 - xy + y = (x - 1) · (x + 1) - y · (x - 1) = (x - 1) · (x - y + 1)
b)   x3 - 5x2 - 3x+ 15 = x2(x - 5) - 3(x - 5) = (x - 5) · (x2 - 3)
Factoring trinomials and polynomials
A quadratic trinomial  ax2 + bx + can be factorized as
ax2 + bx + c = a·[x2 + (b/a)·x + c/a] = a·(x - x1)(x - x2),   where x1 + x2 = b/a  and  x1· x2 = c/a
That means, to factor a quadratic trinomial we should find such a pair of numbers x1 and x2 whose sum equals b/a and whose product equals c/a.
Examples:   a)  x2 - x - 12 =  x2 - 4x + 3x - 12 = x(x - 4) + 3(x - 4) = (x - 4) · (x + 3)
b)  2x2 - 5x + 3 = 2(x2 - 5/2x + 3/2) = 2(x2 - 3/2x - x + 3/2) = 2[x(x - 3/2) - (x - 3/2)] =
= 2(x - 3/2)(x - 1) = (2x - 3)(x - 1)
c)  x3 - 3x + 2 = x3 - 4x + x + 2 = x (x2 - 4) + (x + 2) =x(x - 2)(x + 2) + (x + 2) =
= (x + 2)[x (x - 2) + 1] = (x + 2)(x2  - 2x+ 1) = (x + 2)(x - 1)(x - 1
or           x3 - 3x + 2 = x3 - x - 2x + 2 = x (x2 - 1) - 2(x - 1) = x(x - 1)(x + 1) - 2 (x - 1) =
= (x - 1)[x(x + 1) - 2] = (x - 1)(x2 + x - 2) = (x - 1)(x2 - x + 2x - 2)
= (x - 1)[x(x - 1) + 2(x - 1)] = (x - 1)(x - 1)(x + 2)
Beginning Algebra Contents C