Question

Which of these represents the factorization of x4−256 ?

Answer 1

Answer:

(x+4)(x-4)(x+4i)(x-4i) (answer D)

Step-by-step explanation:

We can re-write the original binomial as a difference of squares noticing that $456=16^2$ and that $x^4=(x^2)^2$

Then we have:

$x^4-256=(x^2)^2-16^2$

Then we can factor this out using the difference of squares factor form:

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

Now, $(x^2-16)$, is itself a difference of squares which we can factor out further:$x^2-16=x^2-4^2=(x+4)(x-4)$

And we can also solve for the binomial: $(x^2+16)$:

$x^2+16=0\\x^2=-16\\x=+/-\sqrt{-16} \\x=+/-i\,\sqrt{16} \\x=+/-4\,i$

then we can write $(x^2+16)=(x+4i)(x-4i)$

Therefore, the final factor form of the original binomial is the product of all factors we found: $(x+4)(x-4)(x+4i)(x-4i)$