Transfer of Ideas Math Example 3

Follow the full solution, then compare it with the other examples linked below.

easy

The factorisation

a^2-b^2=(a-b)(a+b)

transfers to factoring

x^4-16

. Apply it.

1
Write $x^4-16 = (x^2)^2-4^2$ . Apply difference of squares: $(x^2-4)(x^2+4)$ .
2
Factor further: $x^2-4=(x-2)(x+2)$ . The factor $x^2+4$ is irreducible over $\mathbb{R}$ .
3
Final: $x^4-16=(x-2)(x+2)(x^2+4)$ .

Answer

x^4-16=(x-2)(x+2)(x^2+4)

The difference-of-squares identity transfers to expressions where

a

and

b

are not just monomials but higher-degree terms. Recognising

x^4 = (x^2)^2

unlocks the pattern.

About Transfer of Ideas

The ability to recognize that a technique or concept from one area of mathematics applies, possibly in adapted form, to a different area.

The idea of completing the square to solve [formula] transfers to converting [formula] to vertex for

The AM-GM inequality [formula] was originally about two positive numbers. Transfer the idea to prove

The proof technique 'assume the hypothesis and derive the conclusion' (direct proof) from logic tran