Answer:
[tex]a^3 + b^3 = -4[/tex]
Step-by-step explanation:
Given
[tex]a + b = 3ab = 2[/tex]
Required
Solve: [tex]a^3 + b^3[/tex]
Rewrite the expression as:
[tex]a^3 + b^3 = a^3 - a^2b +ab^2 + a^2b - ab^2 + b^3[/tex]
Factorize
[tex]a^3 + b^3 = a(a^2 - ab +b^2) + b(a^2 - ab + b^2)[/tex]
[tex]a^3 + b^3 = (a + b)(a^2 - ab + b^2)[/tex]
Rewrite as:
[tex]a^3 + b^3 = (a + b)(a^2 + b^2- ab)[/tex]
[tex]a^3 + b^3 = (a + b)(a^2 + b^2+2ab - 3ab)[/tex]
Factorize
[tex]a^3 + b^3 = (a + b)((a + b)^2 - 3ab)[/tex]
Open brackets
[tex]a^3 + b^3 = (a + b)^3 - 3ab(a + b)[/tex]
Given that:
[tex]a + b = 3ab = 2[/tex]
This means that:
[tex]a + b = 2[/tex] and [tex]3ab = 2[/tex]
So, we have:
[tex]a^3 + b^3 = (a + b)^3 - 3ab(a + b)[/tex]
[tex]a^3 + b^3 = 2^3 - 3 * 2 * 2[/tex]
[tex]a^3 + b^3 = 8 - 12[/tex]
[tex]a^3 + b^3 = -4[/tex]
