# 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 · (2x3 − 3x2 + 4x − 2) =

= 6x3 − 9x2 + 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.

3x2 · (2x3 − 3x2 + 4x − 2) =

= 6x5 − 9x4 + 12x3 − 6x2

### Multiplication of Polynomials

P(x) = 2x2 − 3    Q(x) = 2x3 − 3x2 + 4x

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

P(x) ·  Q(x) = (2x2 − 3) · (2x3 − 3x2 + 4x) =

= 4x5 − 6x4 + 8x3 − 6x3 + 9x2 − 12x =

Add the monomials of the same degree:

= 4x5 − 6x4 + 2x3 + 9x2 − 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) = 3x4 + 5x3 − 2x + 3 and Q(x) = 2x2 − x + 3

P(x) · Q(x) = (3x4 + 5x3 − 2x + 3) · (2x2 − x + 3) =

= 6x6 − 3x5 + 9x4 + 10x5 − 5x4 + 15x3

− 4x3 + 2x2 − 6x + 6x2 − 3x + 9 =

= 6x6 + 7x5 + 4x4 + 11x3 + 8x2 − 9x + 9