Question

(i) 4x⁴ + y⁴ = (2x² + y² + 2xy) (2x² + y² – 2xy)

(ii) x² – 7x – 8 = (x + 8) (x – 1)

Options

• Only (i)

• Only (ii)

• Both (i) and (ii)

• Neither (i) nor (ii)

Answer 1

Only (i)

Explanation:

Let's analyze each expression for factorization:

(i) 4x⁴ + y⁴

We can rewrite it as:

(2x²)² + (y²)² = (2x²)² + (y²)² + 2(2x²)(y²) – 2(2x²)(y²) 

(2x²)² + (y²)² = (2x² + y²)² – (2xy)²

This is a difference of squares form a² – b² = (a + b)(a – b), with (a = 2x² + y²) and (b = 2xy). 

So, 4x⁴ + y⁴ = (2x² + y² + 2xy)(2x² + y² – 2xy).

(ii) x² – 7x – 8

Let’s check the factorization:

(x + 8)(x – 1) = x² – x + 8x – 8

(x + 8)(x – 1) = x² + 7x – 8

This does not match the original x² – 7x – 8.

The signs in the middle terms differ.

Instead, the correct factorization for x² – 7x – 8 would be:

(x – 8)(x + 1) = x² + x – 8x – 8

(x – 8)(x + 1) = x² – 7x – 8

Therefore, the given factorization for (ii) is incorrect.