Question

Find all zeros of the function
f(y) = y4 – 625

Answer 1

Well it kind of depends on what class you're in.  If you haven't covered complex numbers yet, this is asking for the real zeros.  If you have, we want all the zeros.  We'll calculate all the zeros; you can eliminate the complex ones if you haven't covered that yet.

We'll do this properly, factoring using the difference of two squares.

$y^4 - 625 = 0$

$y^4 - 25^2 = 0$

$(y^2 - 25)(y^2 + 25) = 0$

$(y^2 - 5^2)(y^2 + 25) = 0$

$(y - 5)(y+5)(y^2 + 25)=0$

In complex numbers we can write -25=(5i)² so

$(y - 5)(y+5)(y^2 - (5i)^2)=0$

$(y - 5)(y+5)(y - 5i)(y + 5i)=0$

We have four factors that multiply to zero, so our roots are:

Answer: 5, -5, 5i, -5i

Drop the last two if you haven't learned about complex numbers yet.