If -1 is one of the zeroes of the polynomial

x
3

+2
x
2

−11x−12
, then the other zeroes are ____ and ____.

Question

If -1 is one of the zeroes of the polynomial

x
    3
   
   +2
    x
    2
   
   −11x−12
  , then the other zeroes are ____ and ____.

Accepted Answer

If -1 is one of the zeroes of the polynomial

x
    3
   
   +2
    x
    2
   
   −11x−12
  , then the other zeroes are -4 and 3.Now, here, we are given a polynomial as

x
    3
   
   +2
    x
    2
   
   −11x−12
  and its one zero is -1. We know that for 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+1
   
  is the factor of

x
    3
   
   +2
    x
    2
   
   −11x−12
 . So, using the division algorithm for polynomials, we will divide

x
    3
   
   +2
    x
    2
   
   −11x−12
  by

(x+1)
  and then proceed as per the division algorithm for polynomials.So, we observe that the quotient is obtained as

(
    x
    2
   
   +x−12)
 . Therefore, we can rewrite the given polynomial as follows,

x
    3
   
   +2
    x
    2
   
   −11x−12=
    
     x+1

x
      2
     
     +x−12
   
 .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+1)

x
          2
         
         +4x−3x−12
       =0

(x+1)(x(x+4)−3(x+4))=0

(x+1)(x+4)(x−3)=0

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

(x+4)=0

⇒x=−4

(x−3)=0

⇒x=3

So, the zeros of

x
    3
   
   +2
    x
    2
   
   −11x−12
  are

−1,−4
  and 3.Therefore, the other zeros of the polynomial are obtained as – 4 and 3 respectively.

If -1 is one of the zeroes of the polynomial $x 3 + 2 x 2 − 11 x − 12$ , then the other zeroes are and .

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If -1 is one of the zeroes of the polynomial x 3 +2 x 2 −11x−12 , then the other zeroes are ____ and ____.