Understanding the Expression
To factorise the expression x7y + xy7, we need to look for common factors in both terms.
- The first term is x^7y.
- The second term is xy^7.
Identifying Common Factors
Both terms share common factors of x and y:
- Common factor: xy
Factoring Out the Common Factor
By factoring out xy, we rewrite the expression:
- x^7y + xy^7 = xy(x^6 + y^6)
Now we have a simpler expression, xy multiplied by the sum of x^6 and y^6.
Using Sum of Cubes
We notice that x^6 + y^6 can be factored further:
- x^6 + y^6 = (x^2)^3 + (y^2)^3
Using the sum of cubes formula, a^3 + b^3 = (a + b)(a^2 - ab + b^2), we can apply it here:
- a = x^2
- b = y^2
Applying the formula:
- (x^2 + y^2)(x^4 - x^2y^2 + y^4)
Final Factored Form
Thus, the complete factorization of x7y + xy7 is:
- xy(x^2 + y^2)(x^4 - x^2y^2 + y^4)
Among the options given, option C correctly represents this factorization:
- Option C: xy(x^2 + y^2)(x + y - x^2y^2)
This matches with our derived factors and confirms that option C is indeed the correct answer.