Math, asked by fabia22, 1 year ago

Heyaaa!!


Factorise X⁴ -64​

Answers

Answered by siddhartharao77
20

Step-by-step explanation:

Given Equation is x⁴ - 64

It can be written as,

= (x²)² - (8)²

∴ a² - b² = (a + b)(a - b)

= (x² + 8)(x² - 8)

Hope it helps!


Swetha02: Nice!!!
Answered by Siddharta7
10

Most people trying to factor this expression would say that it is irreducible, that is, it cannot factor at all. I will show you a trick that consist of including two terms that not appear in the original expression.

x4 + 64 = x4 + 16x2 - 16x2 + 64

The two terms 16x2 and -16x2 appear by considering the square root of x4 and by dividing the original constant by the leading exponent (64 ÷ 4 = 16). Then, you change the position of the negative term to the last position:

x4 + 16x2 - 16x2 + 64 = x4 + 16x2 + 64 - 16x2

The first three terms form a perfect square trinomial, which can be easily factored. The last term can be rewritten considering the square root concept.

x4 + 16x2 + 64 - 16x2

= (x2 + 8)(x2 + 8) - (4x)2

= (x2 + 8)2 - (4x)2

The preceding expression is a difference of two sqaures. Considering the pattern for this type of factorizaion (a2 - b2), a equals x2 + 8 while b = 4x. So, knowing that a2 - b2 = (a + b)(a - b), we have

(x2 + 8)2 - (4x)2 = (x2 + 8 + 4x)(x2 + 8 - 4x),

which gives the solution to the exercise.

I'll hope that this helping tool would lead you to solve any similar type of factorization exercises.

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