This section starts with a review of Polynomials and goes on to cover multiplication and division of Polynomials and concludes with the introduction of some special polynomials products.

Section 3: Objectives

Students will:

1. be able to solve problems involving multiplication and devisions of polynomials.

Topic 3.1  Review.

Let's see some of the definitions:

Factor of the number: When two or more numbers are multiplied, each of the numbers is called a factor of the product.

For example, in the product 5 · 11 = 55, 5 and 11 are factors of 55.

Coefficient: Each factor is the coefficient of the product of other factors.

For example, In a term 3xy,

• 3 is a coefficient of xy
  
• x is a coefficient of 3y
  
• y is a coefficient of 3x
  
• xy is a coefficient of 3
  

Generally, the numerical part of a term is called its numerical term of its coefficient.

Thus in the term 3XY, 3 is the numerical coefficient.

Exponent: Sometimes, the products are written as powers.

For example, 4 · 4 · 4 is written as 4³
4 · 4 is written as 4²
a · a · a is written as a³

In a³, 3 is called the exponent or power and 'a' the base; the exponent 3 tells the number of times the base 'a' occurs as a factor in the product.

Monomial: A monomial is a term, which is either a number or a variable with positive integral index or an indicated product of a number and one or more variables.

Examples:

1. 7 is a monomial since it is number
   
2. p is a monomial since it is variable
   
3. 7p is a monomial since it is an indicated product of a number 7 and P
   
4. 7pq is also a monomial since it is a product of 5 and a variable 'pq'.
   ¾ x² y³ is also a monomial
   

Polynomial: A polynomial is an indicated sum of monomials.

Examples:

1. 3x + 7
   
2. (4/7)x² + 6x - 8

Example:

2x + 3, it has two terms namely 2x and 3. The degree of
2x is 1. And the degree of 3 is 0. The greatest of the two degrees is 1.

1. A and B are two polynomials. By adding them we get (A+B), which is also a polynomial. Hence the set of polynomials has Closure Property.
   
2. A + B = B + A (Commutative property)
   
3. (A + B) + C = A + (B + C) (Associative Property)
   
4. The zero polynomial is the identity element under addition.
   
5. If 'A' is a polynomial. Its additive inverse is -A. thus every polynomial has an additive inverse.
   

If A = 3x³ + 4x² - x - 1
   B = 4x³ -3 x²+ 4x + 5

Find A + B and B + A

A + B  = (3x³ + 4x² - x - 1) + (4 x³ - 3 x²+ 4x + 5)
          = (3 + 4) x³ + (4 - 3) x² + (- 1 + 4) x + (- 1 + 5)
          = 7x² + x²+ 3x + 4

B + A = ( 4x³ - 3x² + 4x + 5) + (3x³ + 4x² - x - 1)
          = (4x³ + 3x³) + (- 3x² + 4x²) + (4x - x) + 5 - 1
          = 7x² + x² + 3x + 4

It can be seen that A + B = B + A