Polynomials. Transformation of Polynomials to the Standart Form

The sum of monomials are called polynomial.

If we all terms ofpolynomial write in the standart form and transform the similar terms, then we will obtain the polyinomial of standart form.

Any integral expression we can transform to the polynomial of standart form -it is purpose of transformation (simplification) of integral expressions.

Example 1. Transform the following expression into the polynomial of standart form :

`3a*5b+3ab+2a*(-4b)+b*b` .

At the beginning we transform into standart form the members of polynomial : `15ab+3ab-8ab+b^2` . When we reduce similar members, we will obtain the polynomial of the standart form `10ab+b^2`.

Example 2. Transform the following expression into the polynomial of standart form :

`(3a+5b-2c)+(2a-b+4c)`.

As before parenthesis is the sign "+", then the parenthesis we can omit, by maintaining the signs of all summand, that are in the parenthesis. Then we will obtain: `3a+5b-2c+2a-b+4c=(3a+2a)+(5b-b)+(-2c+4c)=5a+4b+2c`.

Example 3. Transform the following expression into the polynomial of standart form :

`(5a^2b+ab^2)-(3a^2b-4ab^2)`.

As before parenthesis is the sign "-", then the parenthesis we can omit, by changing of all summand, that are in the parenthesis. Then we will obtain: `5a^2b+ab^2-3a^2b+4ab^2=(5a^2b-3a^2b)+(ab^2+4ab^2)=2a^2b+5ab^2` .

Example 4. Transform the given expression into the polynomial of standart form :

`4x^2(x-0.5x^2+3)`.

The product of monomial and polynomial according to the distributive law equals the sum of products of this monomial and each member of the polynomial: `4x^2(x-0.5x^2+3)=4x^2*x-4x^2*0.5x^2+4x^2*3=4x^3-2x^4+12x^2` .

Example 5. Transform the following expression into the polynomial of standart form :

`(2x^2y+3xy^2)(2x+3y+1)`.

`(2x^2y+3xy^2)*(2x+3y+1)=2x^2y*(2x+3y+1)+3xy^2(2x+3y+1)=`

`=(4x^3y+6x^2y^2+2x^2y)+(6x^2y^2+9xy^3+3xy^2)=4x^3y+6x^2y^2+2x^2y+6x^2y^2+9xy^3+3xy^2`

Let′s reduce similar members. We obtain: `4x^3y+12x^2y^2+2x^2y+3xy^2+9xy^3`.