Question

1. $r^4 - s^4 = (r^2 + s^2)(r + s)(r - s)$

Explanation:

Step1: Recognize difference of squares

We know that \(a^2 - b^2=(a + b)(a - b)\). For \(r^4 - s^4\), we can rewrite \(r^4=(r^2)^2\) and \(s^4=(s^2)^2\), so \(r^4 - s^4=(r^2)^2-(s^2)^2\).
Using the difference of squares formula, we get \((r^2 + s^2)(r^2 - s^2)\).

Step2: Factor \(r^2 - s^2\)

Now, \(r^2 - s^2\) is also a difference of squares. Applying the difference of squares formula again, where \(a = r\) and \(b = s\), we have \(r^2 - s^2=(r + s)(r - s)\).

Step3: Combine the factors

Substituting \(r^2 - s^2=(r + s)(r - s)\) back into \((r^2 + s^2)(r^2 - s^2)\), we get \((r^2 + s^2)(r + s)(r - s)\).

Answer:

The equation \(r^4 - s^4=(r^2 + s^2)(r + s)(r - s)\) is verified by factoring using the difference of squares formula twice.