Math, asked by vaibhavimishra9164, 9 months ago

When x³-2x²+ax=b is divided by x²-2x-3, the remainder is x-6. The value of a and b respectively
A. -2, -6
B. 2 and -6
C. -2 and 6
D. 2 and 6

Answers

Answered by soniyash9494
4

Answer:

answer is a. -2, -6

I hope this helps

Answered by nikitasingh79
7

Given: When x³ - 2x² + ax = b is divided by x² - 2x - 3, the remainder is x - 6.  

 

To find : The value of a and b

 

Solution :  

Let p(x) = x³ - 2x² + ax - b

q (x) = x² - 2x - 3

r (x) = x - 6

 

Therefore,

f(x) = p (x) – r (x)

f(x) = x³ - 2x² + ax - b - (x - 6)

f(x) = x³ - 2x² + ax - b - x + 6

f(x) = x³ - 2x² + ax - x - b + 6

f(x) = x³ - 2x² + (a - 1) x - (b - 6)

 

q(x) = x² - 2x - 3

By middle term splitting :  

= x² - 3x + x - 3

= x(x - 3) + 1(x - 3)

= (x + 1) (x - 3)

Thus,  (x + 1) and (x - 3) are factor of f (x)

On putting f(-1) in f(x) :  

f(-1) = 0

f(x) = x³ - 2x² + (a - 1) x - (b - 6)

f(x) = (-1)³ - 2(-1)² + (a - 1) (-1) - (b - 6)

= - 1 - 2 × 1 - a + 1 - b + 6 = 0

= - 1 - 2 - a + 1 - b + 6 = 0

= - 3 + 1 + 6 - a - b = 0

= 4 - a - b = 0

a + b = 4

a = 4 - b ……………(1)

On putting f (3)  in f(x) :  

f (3) = 0

f(x) = x³ - 2x² + (a - 1) x - (b - 6)

= 3³ - 2 (3)² + (a -1) 3 - b + 6 = 0

= 27 - 2 × 9 + 3a - 3 - b + 6 = 0

= 27 - 18 - 3 + 6 + 3a - b = 0

= 27 - 18 + 3 + 3a - b = 0

= 27 - 15 + 3a - b = 0

= 12 + 3a - b = 0

= 3a - b = - 12 ………….(2)

put the value of a from eq 1 in eq 2 :

= 3(4 - b) - b = - 12

= 12 - 3b - b = - 12

= - 4b = - 12 - 12

= - 4b = - 24

= b = 24/4

b = 6

Put this value of b in eq 1 :  

a = 4 - b

a = 4 - 6

a = - 2

Hence the value of a is - 2 and b is  6 .

Among the given options option (C) - 2 and 6 is correct.

HOPE THIS ANSWER WILL HELP YOU…..

 

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