Notes On Factorization of Polynomials Using Algebraic Identities - CBSE Class 9 Maths
Factorisation 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. Algebraic Identities 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)

#### Summary

Factorisation 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. Algebraic Identities 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)

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