Question

Factorise by splitting the middle term to get two perfect squares. 

x⁴ – 5x² + 4

Answer 1

To factorise the expression x⁴ – 5x² + 4 by splitting the middle term and obtaining two perfect squares, we can proceed as follows:

Step 1: Make a substitution

Let y = x², so the expression becomes:

y² – 5y + 4

Step 2: Factorise the quadratic expression

We need to factor y² – 5y + 4. 

We look for two numbers that multiply to 4 and add up to –5. 

These numbers are –1 and –4:

y² – 5y + 4 = (y – 1) (y – 4)

Step 3: Substitute back y = x²

Now, replace y back with x²:

(x² – 1) (x² – 4)

Step 4: Factor each term as a difference of squares

Both terms are differences of squares, so we can factor them further:

x² – 1 = (x – 1) (x + 1) 

x² – 4 = (x – 2) (x + 2)

Thus, the factorised form of the expression x⁴ – 5x² + 4 is (x – 1)(x + 1)(x – 2) (x + 2)

This is the required factorisation using the method of splitting the middle term to get two perfect squares.