Question

If \((x + y) = 3, xy = 2\), then what is the value of \(x ^3+ y^3 :\)

• A. 6
• B. 7
• C. 8
• D. 9

Answer 1

The Correct Option is D

To find the value of \(x^3 + y^3\) given that \(x + y = 3\) and \(xy = 2\), we will use the identity for the sum of cubes:

The identity is:

\(x^3 + y^3 = (x + y)(x^2 - xy + y^2)\)

We know:

• \(x + y = 3\)
• \(xy = 2\)

First, calculate \(x^2 + y^2\) using the identity: 

\((x+y)^2 = x^2 + 2xy + y^2\)

Substituting the given values:

\(3^2 = x^2 + 2 \times 2 + y^2\)

\(9 = x^2 + 4 + y^2\)

Simplifying gives us:

\(x^2 + y^2 = 5\)

Now substitute back into the formula for \(x^3 + y^3\):

\(x^3 + y^3 = (x + y)(x^2 - xy + y^2)\)

\(x^3 + y^3 = 3(x^2 + y^2 - xy)\)

\(x^3 + y^3 = 3(5 - 2)\)

\(x^3 + y^3 = 3 \times 3\)

\(x^3 + y^3 = 9\)

Therefore, the value of \(x^3 + y^3\) is 9, which matches the correct answer.