How do you factor completely #a^2 - 2ab - 15b^2#?

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Answer 1 · 2016-04-18T06:09:31

#a^2-2ab-15b^2 = (a-5b)(a+3b)#

Since this is a homogeneous polynomial of degree #2#, you can treat it like a quadratic in one variable.

Find a pair of factors of #15# which differ by #2#. The pair #5, 3# works, hence:

#a^2-2ab-15b^2 = (a-5b)(a+3b)#

Alternatively, you can complete the square and use the difference of squares identity:

#A^2-B^2=(A-B)(A+B)#

with #A=(a-b)# and #B=4b# as follows:

#a^2-2ab-15b^2#

#=(a-b)^2-b^2-15b^2#

#=(a-b)^2-(4b)^2#

#=((a-b)-4b)((a-b)+4b)#

#=(a-5b)(a+3b)#