Question 1209398
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Let a and b be complex numbers.  If a + b = 4 and a^2 + b^2 = 6 + ab, then what is a^3 + b^3?

{{{system(highlight(matrix(2,6, a + b, "=", 4, "------", eq,  "(i)", a^2 + b^2, "=", 6 + ab, "------", eq, "(ii)")))}}}

<font color = blue><font size = 4><b>a<sup>3</sup> + b<sup>3</sup> = (a + b)(a<sup>2</sup> - ab + b<sup>2</sup>) ---- Applying the sum-of-cubes postulate/sum-of-cubes theorem/algebraic identity
a<sup>3</sup> + b<sup>3</sup> = (a + b)(a<sup>2</sup> + b<sup>2</sup> - ab) 
a<sup>3</sup> + b<sup>3</sup> = 4(6 + ab - ab) ----- Substituting 4 for a + b, and 6 + ab for a<sup>2</sup> + b<sup>2</sup></font></font>
<font color = green><font size = 5>a<sup>3</sup> + b<sup>3</sup> = 4(6) = 24</font></font></b></pre>