Question

If the polynomial $f(x)=ax^{3}+bx-c$ is divisible by the polynomial $g(x)=x^{2}+bx+c$,then c=
(a) b
(b) 2b
(c) $2b^{2}$
(d) -2b

Answer 1

SOLUTION :  

The correct option is (c) : 2b².

If f(x) is divisible by g(x), then remainder will be zero. So, to find the values, find the remainder and put it equal to zero to get the values .

Given : f(x) = ax³ + bx + c and g(x) = x² + bx + c  

DIVISION PROCESS IS IN THE ATTACHMENT .

We get remainder r(x) = bx - acx + ab²x + abc - c

bx - acx + ab²x + abc - c

= x( b - ac + ab²) + c ( ab - 1)

since f(x) is exactly divisible by g(x) , therefore remainder should be zero.

So put   x( b - ac + ab² ) + c ( ab - 1) = 0…………(1)

c ( ab - 1) = 0

ab - 1 = 0  , c≠0

ab = 1

In eq 1 , the condition is true for all value of x . So Put x = 1  

x( b - ac + ab² ) + c ( ab - 1) = 0

b - ac + ab² = 0

b + ab² - ac = 0

b(1 + ab) - ac = 0

On putting ab = 1 and a = 1/b  

b(1 + 1/b × b ) - 1/b × c = 0

b(1 +1 ) - c/b = 0

2b - c/b = 0

-c/b = - 2b  

c/b = 2b  

c = 2b × b  

c = 2b²

Hence, the value of c is 2b² .

HOPE THIS ANSWER WILL HELP YOU..

Answer 2

Answer:

Step-by-step explanation: