Question

On dividing x 3 − 6 x 2 + 5 x + 2 by a polynomial g ( x ) , the quotient and the remainder are x − 1 and x + 1 , respectively. Find g ( x ) .

Answer 1

Answer:

Step-by-step explanation: we know that dividend=divisor*quotient+remainder

By Remainder theorem,

p(x)=g(x)q(x)+r(x)

We have, p(x)=x  

3

−3x  

2

+x+2,q(x)=x−2 and r(x)=−2x+4

∴x  

3

−3x  

2

+x+2=g(x)(x−2)+(−2x+4)

⇒x  

3

−3x  

2

+x+2+2x−4=g(x)(x−2)

⇒g(x)=  

(x−2)

x  

3

−3x  

2

+3x−2

​  

=  

(x−2)

x  

3

−2x  

2

−x  

2

+2x+x−2

​  

 

                                         

=  

(x−2)

[x  

2

(x−2)−x(x−2)+1(x−2)]

​  

 

=  

(x−2)

(x  

2

−x+1)(x−2)

​

=x  

2

−x+1

∴g(x)=x  

2

−x+1