Certainly! We are given the equations:
[tex]\[ x + y = 4 \][/tex]
[tex]\[ xy = 5 \][/tex]
We need to find the value of [tex]\( x^3 + y^3 \)[/tex].
To solve for [tex]\( x^3 + y^3 \)[/tex], we can use the identity involving the sum and product of cubes for two variables [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x^3 + y^3 = (x + y)^3 - 3xy(x + y) \][/tex]
Now, let's break this down step-by-step:
1. Calculate [tex]\( (x + y)^3 \)[/tex]:
[tex]\[
(x + y)^3 = (4)^3 = 64
\][/tex]
2. Calculate [tex]\( 3xy(x + y) \)[/tex]:
[tex]\[
3xy(x + y) = 3 \cdot 5 \cdot 4 = 60
\][/tex]
3. Subtract [tex]\( 3xy(x + y) \)[/tex] from [tex]\( (x + y)^3 \)[/tex]:
[tex]\[
x^3 + y^3 = 64 - 60 = 4
\][/tex]
Therefore, the value of [tex]\( x^3 + y^3 \)[/tex] is:
[tex]\[ \boxed{4} \][/tex]
