We already know that x² - y² = (x - y) (x + y). Further, we have verified that x^3 - y^3 = (x - y) (x² + xy + y²). Class 9

We already know that x² - y² = (x - y) (x + y). Further, we have verified that x^3 - y^3 = (x - y) (x² + xy + y²). Class 9

Question 1.

We already know that x² - y² = (x - y) (x + y). Further, we have verified that x³ - y³ = (x - y) (x² + xy + y²). Observe that x - y is a common factor of x² - y² and x³ - y³. Do you think x - y is also a factor of x⁴ - y⁴ ? Note that x⁴ - y⁴ = (x)² - (y)² = (x² + y²). Can you see how x - y is a factor of x⁴ - y⁴? How about x⁵ - y⁵? Does this also have x - y as a factor? Class 9

Answer:

x⁴ - y⁴ = (x²)² - (y²)²

= (x² - y²) (x² + y²)

= (x - y) (x + y) (x² + y²)

Thus, x - y is also a factor of x⁴ - y⁴.

x⁵ - y⁵ = (x - y) (x⁴ + x³y + x²y² + xy³ + y⁴)

Thus, x - y is also a factor of x⁵ - y⁵.

[Note: In general, x - y is a factor of xⁿ - yⁿ where n is a natural number.]