How do you factor −4x4y4+36?
2 Answers
We first rewrite into
Explanation:
For the moment we put the first
Then we can see that
Putting this al together we get:
This cannot be further factorized without using radicals.
−4x4y4+36
=−4(x2y2−3)(x2y2+3)
=−4(xy−√3)(xy+√3)(x2y2+3)
=−4(xy−√3)(xy+√3)(xy−√3i)(xy+√3i)
Explanation:
I will use the difference of squares identity a few times, so here it is:
a2−b2=(a−b)(a+b)
Since I would like to keep the higher degree terms on the left, I choose to separate out a factor of
−4x4y4+36
=−4(x4y4−9)
=−4((x2y2)2−32)
=−4(x2y2−3)(x2y2+3)
If we allow irrational coefficients we can go a little further:
=−4((xy)2−(√3)2)(x2y2+3)
=−4(xy−√3)(xy+√3)(x2y2+3)
If we allow Complex coefficients we can get a little further still:
=−4(xy−√3)(xy+√3)((xy)2−(√3i)2)
=−4(xy−√3)(xy+√3)(xy−√3i)(xy+√3i)
