Question

Prove the identity x^3 – y^6 = (x – y^2)(x^2 + xy^2 + y^4).

A) Factoring by Grouping
B) Difference of Cubes Formula
C) Sum of Cubes Formula
D) Quadratic Formula

Answer 1

The identity x^3 – y^6 = (x – y^2)(x^2 + xy^2 + y^4) is proven by applying the Difference of Cubes Formula, with a = x and b = y^2, resulting in the desired expression.

Step-by-step explanation:

To prove the identity x^3 – y^6 = (x – y^2)(x^2 + xy^2 + y^4), we can use the Difference of Cubes Formula. The formula states that a^3 – b^3 = (a – b)(a^2 + ab + b^2). Applying this to our given expression with a = x and b = y^2, we have:

x^3 - y^6 = x^3 - (y^2)^3

Now, applying the Difference of Cubes formula:

x^3 - (y^2)^3 = (x - y^2)(x^2 + x(y^2) + (y^2)^2)

Which simplifies to:

(x - y^2)(x^2 + xy^2 + y^4)

Thus, the expression on the right matches the right-hand side of our original identity, confirming that the identity holds true.