1. CLASS X Division Algorithm for Polynomials
By :
N. A. Chandra Sekaran
K.V. Nagarcoil
Chennai Region

2. RECAP : Polynomial
A polynomial in ‘x’ is defined as p(x)= anxn + an-1xn-1+……..+ a1x + a0 where an,an-1,…….a1,a0 are real numbers and n is a non negative integer. Eg. p(x)=3x-6 g(x)=2x2+3x-4 f(x)=x3-7x
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

3. Euclid’s Division Lemma
Let ‘a’ and ‘b’ be two positive integers then there exists unique integers ‘q’ and ‘r’ such that a = bq + r , 0 ≤ r < b
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

4. Division Algorithm
If f(x) and g(x) are any two polynomials with g(x) ≠ 0, then there exist unique polynomials q(x) and r(x) such that f(x) = g(x) . q(x) + r(x) where r(x) =0 or degree of r(x) < degree of g(x)
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

5. Dividing Polynomials
Long division of polynomials is similar to long division of whole numbers.
When you divide two polynomials you can check the answer using the following:
dividend = (quotient . divisor) + remainder
The result is written in the form:
quotient +
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Dividing Polynomials

6. Example: Divide & Check
Example: Divide x2 + 3x – 2 by x – 1 and check the answer. x 1.
+ 2 2.
x x 3. 2x
– 2
2x + 2 4.
– 4 5.
remainder 6.
Answer: x + 2 +
– 4
Check:
(x + 2)
quotient
(x + 1)
divisor
+ (– 4)
remainder
= x2 + 3x – 2
dividend
correct
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Example: Divide & Check

7. Example: Divide & Check
Example: Divide 4x + 2x3 – 1 by 2x – 2 and check the answer. x2
+ x
+ 3
Write the terms of the dividend in descending order.
2x3 – 2x2
Since there is no x2 term in the dividend, add 0x2 as a placeholder.
2x2
+ 4x
2x2 – 2x 1. 2. 6x
– 1 4. 3.
6x – 6 5 5.
Answer: x2 + x + 3 5 6. 7. 8.
Check: (x2 + x + 3)(2x – 2) = 4x + 2x3 – 1 9.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Example: Divide & Check

8. Example: Division With Zero Remainder
Example: Divide x2 – 5x + 6 by x – 2. x
– 3
x2 – 2x
– 3x
+ 6
– 3x + 6
Answer: x – 3 with no remainder.
Check: (x – 2)(x – 3)
= x2 – 5x + 6
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Example: Division With Zero Remainder

9. Example: Division With Nonzero Remainder
Example: Divide x3 + 3x2 – 2x + 2 by x + 3 and check the answer. x2
+ 0x
– 2
Note: the first subtraction eliminated two terms from the dividend.
x x2
0x2
– 2x
+ 2
– 2x – 6
Therefore, the quotient skips a term. 8
Answer: x2 – 2 + 8
Check: (x + 3)(x2 – 2) + 8
= x3 + 3x2 – 2x + 2
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Example: Division With Nonzero Remainder

10. RIDDLES
If you had a pizza with crust thickness ‘a’ and radius ‘z’, what is the volume of the pizza?
pi * z * z * a = pizza
When Deepak was six years old he hammered a nail into his favorite tree to mark his height. Five years later at age eleven, Deepak returned to see how much higher the nail was. If the tree grew by ten inches each year, how much higher would the nail be?
The nail would be at the same height as the tree grows at the top.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

11. 110 kms palindrome number is 73037
If 5+3+2=151022
9+2+4=183652
8+6+3=482466
5+4+5= then =?
143547
Today my car meter reads as kms which is palindrome. How many minimum kms further should be traveled so the car meter finds another palindrome?
110 kms palindrome number is 73037
Can you arrange four nines to make it equal to 100. Hint : use any two mathematical symbols.
/ 9 = 100
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

12. Thank You 
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.