Question

Factorize : X^4+ 9x^2 + 81​

Answer 1

Answer:In the given expression x  

4

+9x  

2

+81, add and subtract 9x  

2

 to make it a perfect square as shown below:

x  

4

+9x  

2

+81=(x  

4

+9x  

2

+81+9x  

2

)−9x  

2

=(x  

4

+18x  

2

+81)−9x  

2

=[(x  

2

)  

2

+(9)  

2

+2(x  

2

)(3)]−9x  

2

 

=(x  

2

+9)  

2

−9x  

2

 

(using the identity (a+b)  

2

=a  

2

+b  

2

+2ab

We also know the identity a  

2

−b  

2

=(a+b)(a−b), therefore,

Using the above identity, the expression (x  

2

+9)  

2

−9x  

2

 can be factorised as follows:

(x  

2

+9)  

2

−9x  

2

=(x  

2

+9)  

2

−(3x)  

2

=(x  

2

+9+3x)(x  

2

+9−3x)

Hence, x  

4

+9x  

2

+81=(x  

2

+9+3x)(x  

2

+9−3x)

Step-by-step explanation: