Question

Factorise the following:

a⁴ + 4b⁴ – 5a²b²

Answer 1

We are asked to factorise the expression:

a⁴ + 4b⁴ – 5a²b²

Step 1: Group terms to recognize a pattern

We can rewrite the expression as:

a⁴ – 5a²b² + 4b²

This looks like a quadratic in terms of a² and b², so we will substitute:

x = a² and y = b²

This transforms the expression into:

x² – 5xy + 4y²

Step 2: Factor the quadratic expression

Now, we factor the quadratic x² – 5xy + 4y².

We need two numbers that multiply to 4y² and add up to –5y.

The two numbers that satisfy this are –4y and –y, because:

–4y × –y = 4y² and –4y + (–y) = –5y

So, we can factor the quadratic as (x – 4y) (x – y)

Step 3: Substitute back x = a² and y = b²

Now, substitute x = a² and y = b² back into the factors:

(a² – 4b²) (a² – b²)

Step 4: Recognize difference of squares

The two terms a² – 4b² and a² – b² are both differences of squares.

We can further factor these:

a² – 4b² = (a – 2b) (a + 2b)

a² – b² = (a – b) (a + b)

Final factorisation:

Thus, the fully factorised form is (a – 2b) (a + 2b) (a – b) (a + b)