Given: [tex](a-1)^3+(a-1)^2=a(a-1)^2[/tex]
Step 1) Expanding (a-1)^3 by using difference of the cube formula
We know, [tex](x-y)^2= x^3 - y^3 -3x^2y+3xy^2[/tex].
Therefore, [tex](a-1)^3 = (a)^3-(1)^2 -3(a)^2(1) +3(x)(1)^2=a^3 -1 -3a^2 +3x[/tex]
Now, expanding (a-1)^2 by using formula of (x-y)^2
[tex](x-y)^2= x^2 - 2xy + y^2.[/tex]
Therefore,
[tex](a-1)^2 = a^2-2a + 1[/tex]
On the left side we get
a^3-3a^2+3a-1+a^2-2a + 1 ( Please put this expression in first box)
Step 2) Combining like terms,
a^3-2a^2+a ( Please put this expression in second box)
Step 3) Factoring out gcf a, we get
a (a^2-2a+1) ( Please put this expression in third box)
Step 4)
Factoring out a^2 -2a+1, (a-1)(a-1).
a(a-1)(a-1) ( Please put this expression in fourth box)
(a-1)(a-1) equals (a-1)^2 because (a-1) is two times there.
Finally, we are given a(a-1)^2 = a(a-1)^2 in last expression on the board.
