Basic rules for multiplication of polynomials

1. The set of polynomials has closure, commutative and associative properties under multiplication. A, B, C are
   three polynomials.

2. Let us multiply A and B to get AB, which also a polynomial. This is, as you know, the closure property.

3. If we multiply A with B or B with A, we get the same product, which is the commutative property.
   Example: A ∗ B = B ∗ A

4. Now, let's see how polynomials exhibit the associative property. Here, we multiply the product of A and B with
   C, and we get same result as when we multiply A with the product of B and C.

   Example: (AB) C = A (BC), which is the associative property.

5. 1 is the identity element under multiplication, which means that when 1 is multiplied with any other number, the
   product is the number itself.

6. Polynomials do not posses multiplicative inverses.

7. We use distributive laws for multiplication of polynomials.

8. The degree of the product of two polynomials is the sum of the degree of the multiplicand and the multiplier.

We use distributive laws for multiplication of polynomials.

Example:

Multiply 2x² - 3x + 4 by 3x² - 2x + 1.

Solution: ( 2x² - 3x + 4 ) ∗ (3x² - 2x + 1)

Take each term in the first polynomial and multiply it with the second polynomial.

= 2x² ( 3x² - 2x + 1 ) - 3x ( 3x² - 2x +1 ) + 4 ( 3x² -2x + 1 )

= ( 6x⁴ - 4x³+ 2x² ) - ( 9x³- 6x² + 3x ) + ( 12x² - 8x + 4 )

= 6x⁴ - 4 x³+ 2x² - 9x³ + 6x² - 3x + 12x² - 8x + 4

= 6x⁴- 13x³+ 20x² - 11x + 4.

Column method for multiplying polynomials

In this method, we write the multiplicand and the multiplier in descending powers of x, arrange one under another,
multiply the multiplicand by every term of the multiplier and add.

2x² - 3x + 4
3x² - 2x + 1
--------------------
6x⁴ - 9x³ + 12x²
        -4x³ + 6x² - 8x
                + 2x² - 3x + 4
-------------------------------------
6x⁴ - 13x³ + 20x² - 11x + 4
------------------------------------------

Note:

1.     The set of all polynomials is closed under multiplication of polynomials, as the product of any two polynomials
       is again a polynomial.

2.     The degree of the product = the sum of the degree of the multiplicand and the multiplier.

Try these questions

Find the products:

1.     ( 3x² - 5x + 4 ) ∗2x

       Answer: = ( 3x²∗2x ) + ( - 5x ∗2x ) + ( 4∗2x )

                       = 6x³ - 10 x² + 8x




2.     ( 6x² - 4x + 3 )∗4 x³

       Answer: = ( 6x²∗4x³ ) + ( - 4x ∗ 4x² ) + ( 3∗4 x² )

                       = 24x⁵ - 16 x⁴+ 12 x³




3.     ( 3x²- 5x + 6 ) ∗( 4x - 3 )

       Answer:                    3 x² - 5x + 6

                                              4x - 3

                           _____________________

       Multiply with 4x: 12 x³- 20 x²+ 24x

       Multiply with -3:             - 9 x²+ 15x - 18

                                   _____________________

                                      12x³ -29x²+ 39x – 18




4.     ( x² + x + 1 ) ( x² - x + 1 )

       Answer:          x² + x + 1

                                x² - x + 1

                                ________

       Multiply with x²: x⁴+ x³+ x²

       Multiply with -x:     - x³ - x² - x

       Multiply with +1:     + x² + x + 1

                                _______________

                                x⁴+ x²+ 1




5.     ( x² - 2x + 1 ) ∗1

       Answer: ( x² - 2x + 1) ∗1 = x² - 2x + 1