Use app×
Join Bloom Tuition
One on One Online Tuition
JEE MAIN 2025 Foundation Course
NEET 2025 Foundation Course
CLASS 12 FOUNDATION COURSE
CLASS 10 FOUNDATION COURSE
CLASS 9 FOUNDATION COURSE
CLASS 8 FOUNDATION COURSE
0 votes
6.5k views
in Polynomials by (48.7k points)
closed by

Divide each of the following polynomials by synthetic division method and also by linear division method. Write the quotient and the remainder. 

i. (2x4 + 3x3 + 4x – 2x2) ÷ (x + 3) 

ii. (x4 – 3x2 – 8) ÷ (x + 4) 

iii. (y3 – 3y2 + 5y – 1) ÷ (y – 1)

1 Answer

0 votes
by (49.4k points)
selected by
 
Best answer

i. Synthetic division: 

(2x4 + 3x3 + 4x – 2x2) ÷ (x + 3) 

Dividend = 2x4 + 3x3 + 4x – 2x2 

∴ Index form = 2x4 + 3x3 – 2x2 + 4x + 0 

∴ Coefficient form of the dividend = (2,3, -2,4,0) 

Divisor = x + 3 

∴ Opposite of + 3 is -3

Coefficient form of quotient = (2, -3, 7, -17) 

∴ Quotient = 2x3 – 3x2 + 7x – 17, 

Remainder = 51

Linear division method:

2x4 + 3x3 + 4x – 2x2 = 2x2 + 3x3 – 2x2 + 4x 

To get the term 2x4, multiply (x + 3) by 2x3 and subtract 6x3

= 2x3(x + 31 – 6x3 + 3x2 – 2x2 + 4x 

= 2x3(x + 3) – 3x3 – 2x2 + 4x 

To get the term – 3x3, multiply (x + 3) by -3xand add 9x2

= 2x3(x + 3) – 3x2(x + 3) + 9x2 – 2x2 + 4x 

= 2x3(x + 3) – 3x2(x + 3) + 7x2 + 4x 

To get the term 7x2, multiply (x + 3) by 7x and subtract 21x, 

= 2x3(x + 3) – 3x2(x + 3) + 7x(x + 3) – 21x + 4x 

= 2x3(x + 3) – 3x2(x + 3) + 7x(x + 3) – 17x 

To get the term -17x, multiply (x + 3) by -17 and add 51,

= 2x3(x + 3) – 3x2(x + 3) + 7x(x + 3) – 17(x + 3) + 51 

= (x + 3) (2x3 – 3x2 + 7x - 17) + 51 

∴ Quotient = 2x3 – 3x2 + 7x – 17, 

Remainder = 51 

ii. Synthetic division: 

(x4 – 3x2 – 8) + (x + 4) 

Dividend = x4 – 3x2 – 8 

∴ Index form = x4 + 0x3 – 3x2 + 0x – 8 

∴ Coefficient form of the dividend = (1,0, -3,0, -8) 

Divisor = x + 4 

∴ Opposite of + 4 is -4

∴ Coefficient form of quotient = (1, -4, 13, -52)

∴ Quotient = x3 – 4x2 + 13x – 52, Remainder = 200 

Linear division method: 

x4 – 3x2 – 8 

To get the term x4, multiply (x + 4) by x3 and subtract 4x3

= x3(x + 4) – 4x3 – 3x2 – 8 

To get the term – 4x3, multiply (x + 4) by -4x2 and add 16x2,

= x3(x + 4) – 4x2(x + 4) + 16x2 – 3x2 – 8 

= x3(x + 4) – 4x2(x + 4) + 13x2 – 8  

To get the term 13x2, multiply (x + 4) by 13x and subtract 52x, 

= x3(x + 4) – 4x2(x + 4) + 13x(x + 4) – 52x – 8 

= x3(x + 4) – 4x2(x + 4) + 13x(x + 4) – 52x – 8 

To get the term -52x, multiply (x + 4) by – 52 and add 208,

= x3(x + 4) – 4x2(x + 4) + 13x(x + 4) – 52(x + 4) + 208 – 8 

= (x + 4) (x3 – 4x2 + 13x – 52) + 200 

∴ Quotient = x3 – 4x2 + 13x – 52, Remainder 200 

iii. Synthetic division:

(y3 – 3y2 + 5y – 1) ÷ (y – 1) 

Dividend = y3 – 3y2 + 5y – 1 

Coefficient form of the dividend = (1, -3, 5, -1) 

Divisor = y – 1 

∴ Opposite of -1 is 1.

∴ Coefficient form of quotient = (1, -2, 3) 

∴ Quotient = y2 – 2y + 3, Remainder = 2

Linear division method:

y3 - 3y2 + 5y – 1 

To get the term y3, multiply (y – 1) by y2 and add y2 

= y2(y – 1) + y2 – 3y2 + 5y – 1 

= y2(y – 1) – 2y2 + 5y – 1 

To get the term -2y2, multiply (y – 1) by -2y and subtract 2y, 

= y2(y – 1) – 2y(y – 1) – 2y + 5y – 1 

= y2(y – 1) – 2y(y – 1) + 3y – 1 

To get the term 3y, multiply (y – 1) by 3 and add 3, 

= y2(y – 1) – 2y(y – 1) + 3(y- 1) + 3 – 1 

= (y – 1)(y2 – 2y + 3) + 2 

∴ Quotient = y2 – 2y + 3, 

Remainder = 2.

Welcome to Sarthaks eConnect: A unique platform where students can interact with teachers/experts/students to get solutions to their queries. Students (upto class 10+2) preparing for All Government Exams, CBSE Board Exam, ICSE Board Exam, State Board Exam, JEE (Mains+Advance) and NEET can ask questions from any subject and get quick answers by subject teachers/ experts/mentors/students.

...