Question

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

x⁴ – 10x² + 9

Answer 1

Given expression: x⁴ – 10x² + 9

We have to find two numbers whose sum is –10 and whose product is 9. These numbers are –9 and –1.

Now, using middle term factorisation method,

x⁴ – 10x² + 9

= x⁴ – 9x² – x² + 9

= x²(x² – 9) – (x² – 9)

= (x² – 9) (x² – 1)

We know that.

a² – b² = (a + b)(a – b)

Hence, x⁴ – 10x² + 9 = (x + 3) (x – 3) (x + 1) (x – 1)

The factorisation result is (x + 3x) (x – 3) (x + 1) (x – 1).