Question

a factor of x^16-y^16 is options are (a)x^2+y^2,(b)x^3+y^3,(c)x^6+y^6 and (d)x^6-y^6

Answer 1

Answer:

(a)

Step-by-step explanation:

x^16-y^16

=(x^8+y^8)(x^8-y^8)

=(x^8+y^8)(x^4+y^4)(x^4-y^4)

=(x^8+y^8)(x^4+y^4)(x^2+y^2)(x^2-y^2)

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Answer 2

Answer:

The given polynomial is;

x^16 - y^16.

We know that ,

a^2 - b^2 = (a - b)•(a + b)

We can use this identity to find the factors of the given polynomial.

=> x^16 - y^16

=> (x^8)^2 - (y^8)^2

=> (x^8 - y^8)•(x^8 + y^8)

=> { (x^4)^2 - (y^4)^2 }•(x^8 + y^8)

=> (x^4 - y^4)•(x^4 + y^4)•(x^8 + y^8)

=> {(x^2)^2 - (y^2)^2}(x^4+y^4)(x^8+y^8)

=> (x^2-y^2)(x^2+y^2)(x^4+y^4)(x^8+y^8)

=> (x-y)(x+y)(x^2+y^2)(x^4+y^4)(x^8+y^8)

Clearly, here we can observe that ,

The factors of the given polynomial are:

x - y

x + y

x^2 + y^2

x^4 + y^4

x^8 + y^8

But , the given opinions are:

a) x^2 + y^2

b) x^3 + y^3

c) x^6 + y^6

d) x^6 - y^6.

Clearly, here the option (a) is the correct answer, as (x^2 + y^2) is a factor of the given polynomial.