How do you factor #(a^2 +1)^2 - 7(a^2 +1) +10#?

2 Answers
Sep 28, 2015

#a_1=1 , a_2=-1, a_3=2, a_4=-2#

Explanation:

Let #(a^2+1) =x#
so the eqn is #x^2-7x+10=0 #
now,
#x^2 -5x-2x+10=0#
#=>x(x-5)-2(x-5)=0#
#=>(x-2)*(x-5)=0#
#=>x-2=0 =>x=2 =>a^2+1=2 => a^2=1#
#=>a=+-1 #
Again,
#=>x-5=0 =>x=5 =>a^2+1=5 => a^2=4#
#=>a=+-2 #

Sep 28, 2015

#(a^2+1)^2-7(a^2+1)+10 = (a+1)(a-1)(a+2)(a-2)#

Explanation:

Let #u = a^2+1#, then the expression is:

#u^2-7u+10# which can be factored:

#(u-2)(u-5)#

Replacing #u#, we get:

#((a^2+1)-2)((a^2+1)-5)#.

We can simplify to get:

#(a^2-1)(a^2-4)#.

Each of these is a difference of squares, so we can finish with:

#(a+1)(a-1)(a+2)(a-2)#