How do you factor #z^4 - 10z^2 + 9#?

1 Answer
Sep 2, 2016

Answer:

#z^4-10z^2+9 = (z-1)(z+1)(z-3)(z+3)#

Explanation:

The difference of square identity can be written:

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

We will use this a couple of times.

Let us treat this as a quadratic in #z^2#. Note that the sum of the coefficients is #0#. That is #1-10+9 = 0#. Hence #((z^2)-1)# is a factor and the other factor must be #((z^2)-9)#...

#z^4-10z^2+9 = (z^2)^2-10(z^2)-9#

#color(white)(z^4-10z^2+9) = (z^2-1)(z^2-9)#

#color(white)(z^4-10z^2+9) = (z^2-1^2)(z^2-3^2)#

#color(white)(z^4-10z^2+9) = (z-1)(z+1)(z-3)(z+3)#