Question

find all the zeros of a polynomial 3x^4+6x^3-2x^2-10x-5if two of the zeros are√5/3 and-√5/3​

Answer 1

Answer:

Two zeroes are  

3

5

​  

 

​  

 and −  

3

5

​  

 

​  

 

So we can write it as, x =  

3

5

​  

 

​  

 and x = −  

3

5

​  

 

​  

 

we get x−  

3

5

​  

 

​  

=0 and x+  

3

5

​  

 

​  

=0

Multiply both the factors we get,

x  

2

−  

3

5

​  

=0

Multiply by 3 we get

3x  

2

−5=0 is the factor of 3x  

4

+6x  

3

−2x  

2

−10x−5

Now divide, 3x  

4

+6x  

3

−2x  

2

−10x−5 by 3x  

2

−5=0 we get,

Quotient is x  

2

+2x+1=0

Compare the equation with quadratic formula,

x  

2

−(Sum  of  root)x+(Product  of  root)=0

⇒Sum of root =2

⇒Product of the root =1

So, we get

⇒x  

2

+x+x+1=0

⇒x(x+1)+1(x+1)=0

⇒x+1=0,x+1=0

⇒x=−1,x=−1

So, our zeroes are −1,−1,  

3

5

​  

 

​  

 and −  

3

5

​  

 

​