If the polynomial

p(x)=

x
4

+a
x
3

−7
x
2

−8x+b

is exactly divisible by

x+2

as well as

x+3
,
then the value of

a
is [[1]] and the value of

b
is [[2]].

Question

If the polynomial

p(x)=

x
      4
     
     +a
      x
      3
     
     −7
      x
      2
     
     −8x+b
   
   is exactly divisible by

x+2
   
   as well as

x+3
   ,
   then the value of

a
   is [[1]] and the value of

b
   is [[2]].

Accepted Answer

If the polynomial

p(x)=

x
      4
     
     +a
      x
      3
     
     −7
      x
      2
     
     −8x+b
   
   is exactly divisible by

x+2
   
   as well as

x+3
   ,
   then the value of

a
   is 2 and the value of

b
   is 12.It is given that the polynomial

p(x)=

x
      4
     
     +a
      x
      3
     
     −7
      x
      2
     
     −8x+b
   
   is exactly divisible by

x+2
   
   as well as

x+3
   .
  We know by Factor theorem that if

x−k
   
   is a factor of the given polynomial

p(x)
  , then

p(k)=0
   and if it is not a factor then

p
    k
   =Remainder.
  Since

x+2
   
   is a factor of the polynomial

p(x)=

x
      4
     
     +a
      x
      3
     
     −7
      x
      2
     
     −8x+b
   ,
   this means that

p
    
     −2
   =0.

⇒p
    
     −2
   =

(−2)
      4
     
     +a
      
       (−2)
      3
     
     −7
      
       (−2)
      2
     
     −8(−2)+b
   =0

⇒16−8a−28+16+b=0

⇒−8a+b+4=0

⇒−8a+b=−4

⇒8a−b=4                          ...(1)

Similarly, if

x+3
   
  is the factor, then

p
    
     −3
   =0.

⇒p
    
     −3
   =

(−3)
      4
     
     +a
      
       (−3)
      3
     
     −7
      
       (−3)
      2
     
     −8(−3)+b
   =0

⇒81−27a−63+24+b=0

⇒−27a+b+42=0

⇒−27a+b=−42

⇒27a−b=42                       ...(2)

Subtracting equation (1) from (2), we get,

27a−b
     −
      
       8a−b
     =42−4

⇒19a=38

⇒a=2

Therefore from equation (1), we get

8×2−b=4

⇒16−b=4

⇒b=12

Therefore, the values of

a=2 and b=12.

If the polynomial $p (x) = x 4 + a x 3 − 7 x 2 − 8 x + b$ is exactly divisible by $x + 2$ as well as $x + 3,$ then the value of $a$ is [[1]] and the value of $b$ is [[2]].

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If the polynomial p(x)= x 4 +a x 3 −7 x 2 −8x+b is exactly divisible by x+2 as well as x+3 , then the value of a is [[1]] and the value of b is [[2]].

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If the polynomial $p (x) = x 4 + a x 3 − 7 x 2 − 8 x + b$ is exactly divisible by $x + 2$ as well as $x + 3,$ then the value of $a$ is [[1]] and the value of $b$ is [[2]].