Four special factorizations occur so often that they are listed here for future reference.} }
SPECIAL FACTORIZATIONS
x - y = (x+y)(x-y) Difference of two squares
x + 2xy + y = (x+y) Perfect square
x - y = (x-y)(x + xy + y) Difference of two cubes
x + y = (x+y)(x - xy + y) Sum of two cubes
A polynomial that cannot be factored is called a prime polynomial.
EXAMPLE
Factor each of the following.
(a) 64p-49q= (8p)-(7q)= (8p+7q)(8p-7q)
(b) x + 36 is a prime polynomial.
(c) x + 12x +36 = (x +6)
(d) 9y-24yz + 16z = (3y-4z)
(e) y - 8 = (y-2)(y + 2y + 4)
(f ) m +125 = m + 5 = (m+5)(m -5m + 25)
(g) 8k-27z = (2k)-(3z) = (2k -3z)(4k +6kz +9z)
CAUTION
In factoring, always look for a common factor first. Since 36x - 4y has a common factor of 4,
36x - 4y = 4(9x - y) = 4(3x + y)(3x - y)
It would be incomplete to factor it as
36x - 4y = (6x + 2y)(6x - 2y)
since each factor can be factored still further. To factor means to factor completely, so that each polynomial factor is prime.
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