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

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.

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.

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.

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

Example: Divide & Check Example: Divide x2 + 3x – 2 by x – 1 and check the answer. x 1. + 2 2. x2 + 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

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) + 5 = 4x + 2x3 – 1 9. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Divide & Check

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

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. x3 + 3x2 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

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.

110 kms palindrome number is 73037 If 5+3+2=151022 9+2+4=183652 8+6+3=482466 5+4+5=202541 then 7+2+5=? 143547 Today my car meter reads as 72927 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. 99 + 9 / 9 = 100 Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

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