# Multiplying Polynomials

### Multiplication of a Number by a Polynomial

It is another** polynomial** that has the same ** degree**. The coefficients are the product of the coefficients of the polynomial and the number.

3 · (2x^{3} − 3x^{2} + 4x − 2) =

= 6x^{3} − 9x^{2} + 12x − 6

### Multiplication of a Monomial by a Polynomial

The monomial is multiplied by each and every one of the monomials that form the polynomial.

3x^{2} · (2x^{3} − 3x^{2} + 4x − 2) =

= 6x^{5} − 9x^{4} + 12x^{3} − 6x^{2}^{}

### Multiplication of Polynomials

P(x) = 2x^{2} − 3 Q(x) = 2x^{3} − 3x^{2} + 4x

Multiply each monomial from the first polynomial by each of the monomials in the second polynomial.

P(x) · Q(x) = (2x^{2} − 3) · (2x^{3} − 3x^{2} + 4x) =

= 4x^{5} − 6x^{4} + 8x^{3} − 6x^{3} + 9x^{2} − 12x =

Add the monomials of the same degree:

= 4x^{5} − 6x^{4} + 2x^{3} + 9x^{2} − 12x

The multiplication of polynomials is another* polynomial* whose

**degree**is the

**sum**of the

**degrees**of

**the polynomials**that are to be

**multiplied**.

The polynomials can also be multiplied as follows:

#### Example

**Multiply the polynomials** using two different methods:

P(x) = 3x^{4} + 5x^{3} − 2x + 3 and Q(x) = 2x^{2} − x + 3

P(x) · Q(x) = (3x^{4} + 5x^{3} − 2x + 3) · (2x^{2} − x + 3) =

= 6x^{6} − 3x^{5} + 9x^{4} + 10x^{5} − 5x^{4} + 15x^{3} −

− 4x^{3} + 2x^{2} − 6x + 6x^{2} − 3x + 9 =

= 6x^{6} + 7x^{5} + 4x^{4} + 11x^{3} + 8x^{2} − 9x + 9