Factoring Polynomials

Factoring a polynomial is to find polynomials whose product is equal to the polynomial.

A prime polynomial is a polynomial that can not be factored as the product of two polynomials with lower degrees.

Remark A prime polynomial in one variable has degree one or two.

A completely factored polynomial is a polynomial that is factored as a product of prime polynomials.

Commonly used tricks in factoring polynomials

$ \diamond$ Factoring out common factor:
Ex. $ 2x^2+x^3=x^2(2+x)$
$ \diamond$ Grouping appropriate terms:
Ex. $ ax+ay+2x+2y= a(x+y)+2(x+y)=(x+y)(a+2)$
$ \diamond$ Factor trinomials: $ x^2+a x+b=(x+x_1)(x+x_2)$ where $ a=x_1+x_2$ and $ b=x_1 x_2$
Ex. $ x^2+3x+2=(x+1)(x+2)$
$ \diamond$ Using the formulas:
$ x^2-y^2=(x+y)(x-y)$ Ex. $ 289x^2y^4-1=(17xy^2+1)(17xy^2-1)$
$ x^2+2xy+y^2=(x+y)^2$ Ex. $ y^2+4y+4=(y+2)^2$
$ x^2-2xy+y^2=(x-y)^2$ Ex. $ y^2-6y+9=(y-3)^2$
$ x^3+y^3=(x+y)(x^2-xy+y^2)$ Ex. $ y^3+8=(y+2)(y^2-2y+4)$
$ x^3-y^3=(x-y)(x^2+xy+y^2)$ Ex. $ y^3-8=(y-2)(y^2+2y+4)$


Department of Mathematics
Last modified: 2005-08-30