On dividing

3
x
3

+
x
2

+2x+5
by a polynomial g(x), the quotient and remainder are (3x – 5) and (9x + 10) respectively, then the value of g(x) is ____.

Question

On dividing

3
    x
    3
   
   +
    x
    2
   
   +2x+5
  by a polynomial g(x), the quotient and remainder are (3x – 5) and (9x + 10) respectively, then the value of g(x) is ____.

Accepted Answer

On dividing

3
    x
    3
   
   +
    x
    2
   
   +2x+5
  by a polynomial g(x), the quotient and remainder are (3x – 5) and (9x + 10) respectively, then the value of g(x) is

x
    2
   
   +2x+1
 .Now, if a polynomial say

f
    x
   
  is divided by another polynomial say

g(x)
  to give the quotient as say

q
    x
   
  and the remainder as

r
    x
   
 , then by the division algorithm for polynomials, we have,

f(x)=q(x)×g(x)+r(x)..........(i)
 The given values of the dividend, quotient and remainder are as follows,

f
      x
     =3
      x
      3
     
     +
      x
      2
     
     +2x+5

q
      x
     =3x−5

r
      x
     =9x+10

Substituting these values in the equation (i) and simplifying, we get,

(3x−5)×g(x)+(9x+10)=3
      x
      3
     
     +
      x
      2
     
     +2x+5

(3x−5)×g(x)=3
      x
      3
     
     +
      x
      2
     
     +2x+5−(9x+10)

(3x−5)×g(x)=3
      x
      3
     
     +
      x
      2
     
     −7x−5

g(x)=
      
       3
        x
        3
       
       +
        x
        2
       
       −7x−5
      
       (3x−5)

So, we will find the value of

g(x)
  by performing long division as follows,So, the quotient obtained after the long division is

x
    2
   
   +2x+1
 .Hence, the value of

g(x)
  is

x
    2
   
   +2x+1
 .

On dividing $3 x 3 + x 2 + 2 x + 5$ by a polynomial g(x), the quotient and remainder are (3x – 5) and (9x + 10) respectively, then the value of g(x) is ____.

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On dividing 3 x 3 + x 2 +2x+5 by a polynomial g(x), the quotient and remainder are (3x – 5) and (9x + 10) respectively, then the value of g(x) is ____.