Question

Factorise the following:

x⁴ – 20x²y² + 64y⁴

Answer 1

Given: x⁴ – 20x²y² + 64y⁴

Step-wise calculation:

1. Recognize the expression as a quadratic in terms of x² and y²:

(x²)² – 20(x²)(y²) + (8y²)²

2. Check if it can be factored as a perfect square or using the quadratic approach:

We try to write it as

(x² – ay²)(x² – by²)

where (a) and (b) satisfy:

a + b = 20

ab = 64

3. Solve for (a) and (b):

Factors of 64 that add to 20 are 16 and 4.

So, a = 16, b = 4.

4. Thus, the factorization becomes:

(x² – 16y²)(x² – 4y²).

5. Further factor using the difference of squares:

x² – 16y² = (x – 4y)(x + 4y)

x² – 4y² = (x – 2y)(x + 2y)