Question

Factorise the following:

x⁴ – 13x² + 36

Answer 1

We are given the expression:

x⁴ – 13x² + 36

Step 1: Substitute for x²

Let’s substitute y = x², transforming the expression into a quadratic form y² – 13y + 36.

Step 2: Factor the quadratic expression

Now, we need to factor y² – 13y + 36.

We are looking for two numbers that multiply to 36 and add up to –13.

The two numbers are –9 and –4, since –9 × –4 = 36 and –9 + (–4) = –13.

Thus, we can factor the quadratic as (y – 9) (y – 4)

Step 3: Substitute back y = x²

Now, substitute y = x² back into the factored form (x² – 9) (x² – 4).

Step 4: Recognize the difference of squares

Both x² – 9 and x² – 4 are differences of squares, which can be factored further:

x² – 9 = (x – 3) (x + 3)

x² – 4 = (x – 2) (x + 2)

Final factorisation:

Thus, the fully factorised form is (x – 3) (x + 3) (x – 2) (x + 2).