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Factorization of Polynomials Using Algebraic Identities

Summary

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HD 4:36 Animated video Lecture for Factorization of Polynomials Using Algebraic Identities

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 + c– 3abc = (a + b + c)(a2 + b2 + c2 – ab – bc – ca)

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Questions & Answers

1 . x3 +1/x3 = 110,then x+1/x =
Given x 3+ 1/x 3 =110
Recall that (x+1/x) 3 = x 3+ 1/x 3 + 3(x+1/x)...
2 . LCM and HCF of two quadratic polynomials are x3-7x+6
Given (x −1) is the HCF
is a factor of both p(x) and q(x)
Given LCM = x 3 7x +6
=
Since both...
4 . If polynomials ax3+3x2-3 and 2x3-5x+a leaves the same remainder
If x2-3x+2 is a factor of poynomial x4-ax3+b then find the value of a and b?
5 . Factorise - 4a2 - 9b2 - 2a - 3b
Consider 4a 2 - 9b 2 - 2a - 3b
= (2a) 2 – (3b) 2 – 2a – 3b
= (2a – 3b)(2a +...