How do you factor by grouping #x^3 + x^2 - x - 1#?

2 Answers
May 25, 2015

#x^3+x^2-x-1#

#=(x^3+x^2)-(x+1)#

#=x^2(x+1)-(x+1)#

#=(x^2-1)(x+1)#

#=(x^2-1^2)(x+1)#

#=(x-1)(x+1)(x+1)#

...using the difference of squares identity

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

Dec 31, 2017

Answer:

Find a useful grouping, then factor.

Answer:  #(x + 1)^2 (x - 1)#

Explanation:

Factor   #x^3+x^2−x−1#

The idea of grouping #x^2# with -#1# looks really tempting
because it's the Difference of Two Squares..

1) Find a useful grouping
# (x^3 - x) + (x^2 - 1)#

2) Factor each group
#x (x^2 - 1) +1 (x^2 - 1)#

3) Factor out #(x^2 - 1)# from each group
#(x^2 - 1)(x + 1)#

4) Factor #(x^2 - 1)# as the Difference of Two Squares
#(x - 1)(x + 1)(x + 1)# #larr# answer

5) You can write it this way if you want:
#(x + 1)^2 (x - 1)# #larr# same answer