Question

The expression x⁴+4 can be factorized as
A. (x²+2x+2)(x²-2x+2)
B. (x²+2x+2)(x²+2x-2)
C. (x²-2x-2)(x²-2x+2)
D. (x²+2)(x²-2)

Answer 1

The given algebraic expression is: x⁴ + 4  

Adding and subtracting 4x² on both sides :

= x⁴ + 4x² + 4 - 4x²

By using an identity, a² + 2ab + b² = (a + b)² :

= {(x²)² + 2 × x² × 2 + 2²} - 4x²

= (x² + 2)² - 4x²

= (x² + 2)² - (2x)²

By using an identity, a² - b² = (a + b) (a - b) :

Here, a = (x² + 2) and b = 2x

= (x² + 2 + 2x) (x² + 2 - 2x)

Hence, the expression x⁴ + 4 can be factorized as (x² + 2 + 2x) (x² + 2 - 2x).

Option (A)  (x² + 2 + 2x) (x² + 2 - 2x) is correct.

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Answer 2

Given equation:

x⁴ + 4

Solution:

In order to factorise it, we need to make perfect squares. Thus, for making so, we should add and subtract "4x²" from the given equation.

x⁴ + 4 + 4x² - 4x²

Using the identity:

(a + b)² = a² + 2ab + b²

=> (x² + 2) - (2x)²

Using the identity:

a² - b² = (a + b)(a - b)

=> (x² + 2 - 2x)(x² + 2 + 2x)

Thus, (A) is correct.