If

5

,−
5

are zeroes of the polynomial

x
4

+4
x
3

−2
x
2

−20x−15
, then the other zeroes are ____ and ____.

Question

If

5
   
   ,−
    5

are zeroes of the polynomial

x
    4
   
   +4
    x
    3
   
   −2
    x
    2
   
   −20x−15
  , then the other zeroes are ____ and ____.

Accepted Answer

If

5
   
   ,−
    5

are zeroes of the polynomial

x
    4
   
   +4
    x
    3
   
   −2
    x
    2
   
   −20x−15
  , then the other zeroes are -1 and -3.We know that form any polynomial say

f(x)
 , if

f(a)
  is a zero, then

(x−a)
  is said to be a factor of

f(x)
 . So, we can say that

x−
      5

and

x+
      5

are the factors of

x
    4
   
   +4
    x
    3
   
   −2
    x
    2
   
   −20x−15
 . Then,

x−
      5

x+
      5

=(
    x
    2
   
   −5)
  will also be a factor of the given polynomial.So, using the division algorithm for polynomials, we will divide

x
    4
   
   +4
    x
    3
   
   −2
    x
    2
   
   −20x−15
  by

(
    x
    2
   
   −5)
 . So, we observe that the quotient is obtained as

(
    x
    2
   
   +4x+3)
 . Therefore, we can rewrite the given polynomial as follows,

x
    4
   
   +4
    x
    3
   
   −2
    x
    2
   
   −20x−15=

x
      2
     
     −5

x
      2
     
     +4x+3
   
 .Now, to find the other zeros, we will set the right side of the above equation to 0 and factorize using the method of middle-term splitting as follows,

x
        2
       
       −5

x
        2
       
       +4x+3
     =0

x
        2
       
       −
        
         (
          5
         
         )
        2

x
        2
       
       +4x+3
     =0

(x+
      5
     
     )(x−
      5
     
     )

x
        2
       
       +3x+x+3
     =0

(x+
      5
     
     )(x−
      5
     
     )(x+1)(x+3)=0

Now, we will use the zero-product property and determine the zeros as follows,

(x+1)=0 or 
      
       x+3
     =0

⇒x=−1 or x=−3

So, the zeros of

x
    4
   
   +4
    x
    3
   
   −2
    x
    2
   
   −20x−15
  are

5
   
   ,−
    5
   
   ,−1
  and

−3
 .Therefore, the other zeros are obtained as – 1 and –3 respectively.

If $5, − 5$ are zeroes of the polynomial $x 4 + 4 x 3 − 2 x 2 − 20 x − 15$ , then the other zeroes are and .

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If 5 ,− 5 are zeroes of the polynomial x 4 +4 x 3 −2 x 2 −20x−15 , then the other zeroes are ____ and ____.