Reminder and Factor Theorem

Question 1

Reminder Theorem

Answer 1

If p(x) is any polynomial and ‘a’ is any real number, then the remainder when p(x) is divided by (x-a) is given by p(a).

Question 2

Factor Theorem

Answer 2

If p(x) is a polynomial of degree n(>= 1) and ‘a’ be any real number such that p(a)=0 then (x-a) is a factor of p(x).

Question 3

Horner’s Process for Synthetic Division of Polynomial

Answer 3

When a polynomial f ( x ) = p 0 x n + p 1 x n − 1 + … + p n − 1 x + p n is divided by a binomial x − α , let the quotient be Q ( x ) and remainder be r . We can find quotient Q ( x ) and remainder r by using Horner’s synthetic division process as explained below. Step 1: Write all the coefficients p 0 , p 1 , p 2 , …, p n of the given polynomial f ( x ) in the order of descending powers of x as in the first row. When any term in f ( x ) (as seen with descending powers of x ) is missing, we write zero for its coefficient. Step 2: Divide the polynomial f ( x ) by ( x − α ) by writing α in the left corner as shown above ( x − α = 0 ⇒ x = α ). Step 3: Write the first term of the third row as q 0 = p 0 , then multiply q 0 by α to get q 0 α and write it under p 1 , as the first element of the second row. Step 4: Add q 0 α to p 1 to get q 1 , the second element of the third row. Step 5: Again multiply q 1 with α to get q 1 α and write q 1 α under p 2 and add q 1 α to p 2 to get q 2 which is the third element of the third row. Step 6: Continue this process till we obtain q n − 1 in the third row. Multiply q n − 1 with α and write q n − 1 α under p n and add q n − 1 α to p n to get r in the third row as shown in previous page. In the above process, the elements of the third row, i.e., q 0 , q 1 , q 2 , …, q n − 1 are the coefficients of the quotient Q ( x ) in the same order of descending powers starting with x n − 1 . ∴ Q ( x ) = q 0 x n − 1 + q 1 x n − 2 + … + q n − 2 x + q n − 1 and the remainder is r , i.e., the last element of the third row