1. Rewrite the cubic equation y3 + 3y2 + y + 1 = 0 if the form of a expressed cubic x2 + px + q = 0
using transformation y = r - a/5, a = 3
Solution
Let y=x-a/3, so y=x-1 (because a=3). Then
y3+3y2+y+1=(x-1)3+3(x-1)2+(x-1)+1
Now, we can expand each individual term.
(x-1)3=(x-1)(x-1)(x-1)=(x2-2x+1)(x-1)=x3-3x2+3x-1
3(x-1)2=3(x2-2x+1)=3x2-6x+3
Now, we can combine the expanded terms together.
(x-1)3+3(x-1)2+(x-1)+1=
x3-3x2+3x-1+3x2-6x+3+(x-1)+1
Collect like terms together, so
=x3-3x2+3x2+3x-6x+x-1+3-1+1
=x3-2x+2=0, and thus the equation is in the desired form.