Expansions

An algebraic equation, which is true for all the values of the variables in it, is called an algebraic identity.

To factorise the polynomials using the algebraic identities, consider the following steps:

    Identify the given polynomial as the right hand side of some algebraic identity.

    Apply the identity to the polynomial.

    Factorise according to the identity.

Algebraic identities are listed as follows:

    (a + b)2 = a2 + 2ab + b2

    (a – b)2 = a2 – 2ab + b2

    (a + b)2 + (a – b)2 = 2(a2 + b2)

    (a + b)2 – (a – b)2 = 4ab

    (a + b)(a – b) = a2 – b2

    (a + b)3 = a3 + b3 + 3ab (a + b)

    (a – b)3 = a3 – b3 – 3ab (a – b)

    (ax + b)(cx + d) = acx2 + (ad + bc)x + bd

    (x + b)(x + d) = x2 + (b + d)x + bd

    (x – b)(x – d) = x2 – (b + d)x + bd

    (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca

    a3 + b3 + c3 – 3abc = (a + b + c)(a2 + b2 + c2 – ab – bc – ca)

    If a + b + c = 0, then a3 + b3 + c3 = 3abc

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