Max@Math Revolution wrote:[GMAT math practice question]

The sum of two numbers is 1 and their product is -1. What is the sum of their cubes?

A. 1
B. 2
C. 3
D. 4
E. 5

We can let a and b be the two numbers. So we have a + b = 1 and ab = -1 and we need to determine the value of a^3 + b^3.

Notice that (a + b)^3 = a^3 + 3a^2*b + 3a*b^2 + b^3. So we have:

1^3 = a^3 + b^3 + 3a^2*b + 3a*b^2

1 = a^3 + b^3 + 3ab(a + b)

1 = a^3 + b^3 + 3(-1)(1)

1 = a^3 + b^3 - 3

4 = a^3 + b^3

Alternate Solution:

Letting a and b denote the numbers, we can use the identity a^3 + b^3 = (a + b)(a^2 - ab + b^2). We already know a + b = 1 and ab = -1, we need to find a^2 + b^2.

Notice that (a + b)^2 = a^2 + 2ab + b^2. Since a + b = 1 and ab = -1, we have

1^2 = a^2 + 2(-1) + b^2

a^2 + b^2 = 1 + 2 = 3.

Now, a^3 + b^3 = (a + b)(a^2 - ab + b^2) = (1)(a^2 + b^2 - ab) = 3 -(-1) = 3 + 1 = 4.

Answer: D