Identity and Remainder Theorem

Identity

1. Identity
   An equation involving variables that holds for all values of the variables.
   
   
2. Various Expressions of Identity
1. An equation that holds true regardless of the value of
2. An equation that holds true for all
3. An equation that holds true for any value of
4. An equation that holds true for every possible value of


3. Properties of Identity
1. is an identity in
   
2. is an identity in
   , ,
3. is an identity in and
   


4. Method of Undetermined Coefficients
   A method used to determine unknown coefficients by utilizing the meaning and properties of identities.
1. Coefficient Comparison Method : Compare the coefficients of like terms on both sides of the equation to determine the values.
2. Substitution Method: Substitute appropriate values into the equation to determine the coefficients.

Remainder Theorem and Factor Theorem

1. Remainder Theorem
   When dividing polynomial by the linear polynomial , the quotient is and the remainder is , resulting in the equation:
   ( is a constant)
   Since this is an identity in , substituting into both sides gives . This is known as the Remainder Theorem.
1. The remainder when dividing polynomial by is .
2. The remainder when dividing polynomial by is .


2. Factor Theorem
   For a polynomial
1. If is divisible by , then
   
   
   has as a factor.
2. If is divisible by (where, , then
   .
3. Synthetic Division
   A method for dividing a polynomial by a linear factor using only the coefficients of the terms to find the quotient and remainder.
   
   Example:
   
   
   

 Solution

Quotient:
Remainder: