If g(x) and h(x) are two polynomials whose product is p(x). This can be written as p(x) = g(x) . h(x). g(x) and h(x) are called the factors of the polynomial p(x).
The process of resolving a given polynomial into factors is called factorisation. A non-zero constant is a factor of every polynomial.
Polynomials can be factorised using algebraic identities.
A polynomial of degree two is called a quadratic polynomial. The identities used to factorise the quadratic polynomials are:
- (a + b)2 = a2 + 2ab + b2
- (a – b)2 = a2 – 2ab + b2
- a2 – b2 = (a + b)(a – b)
- (x + a)(x + b) = x2 + (a + b)x + ab
- (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
A polynomial of degree three is called a cubic polynomial. The algebraic identities used in factorising a cubic polynomial are:
- (a + b)3 = a3 + b3 + 3ab (a + b)
- (a – b)3 = a3 – b3 – 3ab (a – b)
- a3 + b3 = (a + b)(a2 – ab + b2)
- a3 – b3 = (a – b)(a2 + ab + b2)
- a3 + b3 + c3 – 3abc = (a + b + c)(a2 + b2 + c2 – ab – bc – ca)