Question on Factorisation using identities

Topic : Factorisation using identities

Problem : Factorise m^4 – 256

Solution :
We know that (a + b)^ 2 = a^2 + 2ab + b^2 …………… (I)
(a – b)^ 2= a^2 – 2ab + b^2………….. (II)
(a + b) (a – b) = a^2 – b^2………… (III)
Let us use these identities for factorisation.

We note m^4 = (m2)^2 and 256 = (16)^ 2
Thus, the given expression fits Identity III.
Therefore, m^4 – 256 = (m2)^2 – (16)^ 2
= (m^2 –16) (m^2 +16) [(using Identity (III)]
Now, (m^2 + 16) cannot be factorised further, but (m^2 –16) is factorisable again as per Identity III.
m^2–16 = m^2 – 4^2
= (m – 4) (m + 4)
Therefore, m^4– 256 = (m – 4) (m + 4) (m^2 +16)