The real numbers $a$ and $b$ satisfy $a - b = 1$ and $a^3 - b^3 = 1.$
(a) Find all possible values of $ab.$
(b) Find all possible values of $a + b.$
(c) Find all possible values of $a$ and $b.$
a - b = 1
a = b + 1
a^3 = (b+1)^3
= b^3 +3b^2 +3b + 1
As mentioned in the question above, a^3 - b^3 = 1
We have proven a^3 = b^3 +3b^2 +3b + 1
So, replacing a^3 with b^3 +3b^2 +3b + 1:
b^3 +3b^2 +3b + 1 - b^3 = 1
Collecting like terms
3b^2 + 3b = 0
3b^2 = -3b
b^2 = -b
b = -1, b = 0
If b = -1:
a-(-1) = 1
a + 1 = 1
a = 0
If b = 0:
a - 0 = 1
a = 1
So if b = -1, a = 0
If b = 0, a = 1
Part (a)
It can be either -1 * 0
Which equals 0
Or 0 * 1
Which equals 0
So the only answer is 0
Part (b)
It can be either -1 + 0
Which equals -1
Or 0 + 1
Which equals 1
So the answers are 1, -1
Part (c)
The possible values for a are 0, 1
The possible values for b are -1, 0