Short Multiplication Formulas

In some cases the transformation of integral expression into the standart form of polynomial realize with the using of identities:

  1. `(a+b)(a-b)=a^2-b^2`
  2. `(a+b)^2=a^2+2ab+b^2`
  3. `(a-b)^2=a^2-2ab+b^2`
  4. `(a+b)(a^2-ab+b^2)=a^3+b^3`
  5. `(a-b)(a^2+ab+b^2)=a^3-b^3`
  6. `(a+b)^3=a^3+3a^2b+3ab^2+b^3`
  7. `(a-b)^3=a^3-3a^2b+3ab^2-b^3`

These identities are called short multiplication formulas; the formula 1- difference of squares; the formulas 2 and 3 - respectively the square of sum and the square of difference, the formulas 4 and 5 - the sum of cubes and difference of cubes, and formulas 6 and 7- the cube of sum and cube of difference.

Example 1. Reduce the following expression to polynomial of standart form :

`(3x^2+4y^3)(3x^2-4y^3)`. Using of formula 1, we obtain: `(3x^2)^2-(4y^3)^2=9x^4-16y^6` .

Example 2. Reduce the following expression to polynomial of standart form :

`(3a^2-5b^3)^2` .

According to formula 3, we find `(3a^2)^2-2*3a^2*5b^3+(5b^3)^2=9a^4-30a^2b^3+25b^6` .

Example 3. Reduce the following expression to polynomial of standart form :

`(3a+1)(9a^2-3a+1)`.

Using of formula 4, we have `(3a)^3+1=27a^3+1`.