Fractoring Polynomials

Special Factorizations

Four special factorizations occur so often that they arelisted 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 primepolynomial.

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 factormeans to factor completely, so that each polynomial factor isprime.