How do you factor #36a^3b^2+66a^2b^3-210ab^4#?

1 Answer
Apr 11, 2017

Answer:

#36a^3b^2+66a^2b^3-210ab^4=color(green)((6ab^2)(3a+5b)(2a-7b))#

Explanation:

Extracting the common factors from each term:
#color(white)("XXX"){: (ul(36),ul(66),ul(210)," | ",6), (ul(a^3),ul(a^2),ul(a)," | ",a), (ul(b^2),ul(b^3),ul(b^4)," | ",b^2) :}#
giving
#color(white)("XXX")(6ab^2)(6a^2+11ab-35b^2)#

Looking for integer (we are optimists) coefficients as factors for #(6a^2+11ab-35b^2)#

#color(white)("XXX"){: (ul("factors of 6"),ul("factors of 35"),ul("difference of cross product")), (1,5,), (ul(6),ul(7),ul(23)), (1,7,), (ul(6),ul(5),ul(37)), (2,5,), (ul(3),ul(7),ul(1)), (2,7,), (ul(3),ul(5),ul(11)) :}#
...and we have found a set that gives us the coefficient of the middle term!

It only remains to figure out the positive/negative signs to give #+11#

#(6ab^2)(2a-7b)(3a+5b)#