Question

Factorise:x6 - 26x3 - 27.
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Answer 1

Answer:

x6 - 78 - 27

x6- 105

x-105

Answer 2

Answer:

Let x ^3 =a, then the equation x^6  −26x^3 −27 becomes a^2 −26a−27

Consider the equation a^2 −26a−27 and factorise it as follows:

a ^2 −26a−27=a^2 −27a+a−27=a(a−27)+1(a−27)=(a−27)(a+1)

Now, substitute the value of a as a=x^3 :

a^2 −26a−27=a^2 −27a+a−27=a(a−27)+1(a−27)=(a−27)(a+1)=(x ^3−27)(x^3 +1)

=(x ^3 −3^ 3  )(x ^3 +1^ 3 )=(x−3)(x^ 2  +3x+9)(x+1)(x^2  −x+1)

(Using identities a^3  +b^3 =(a+b)(a ^2  +b^ 2 −ab) and a^3 −b^3 =(a−b)(a^ 2 +b^ 2 +ab))

Hence, x^6 −26x^3 −27=(x−3)(x^2 +3x+9)(x+1)(x^2 −x+1)

THANKS!