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Suppose that for some [tex]a,b,c[/tex] we have [tex]a+b+c = 1[/tex], [tex]ab+ac+bc = abc = -4[/tex]. What is [tex] a^3+b^3+c^3?[/tex]

Answer :

LammettHash

Consider the cubic polynomial,

[tex](x+a)(x+b)(x+c)[/tex]

Expanding this gives

[tex]x^3+(a+b+c)x^2+(ab+ac+bc)x+abc=x^3+x^2-4x-4[/tex]

We can factor this by grouping,

[tex]x^3+x^2-4x-4=x^2(x+1)-4(x+1)=(x^2-4)(x+1)=(x-2)(x+2)(x+1)[/tex]

Then letting [tex]a=-2[/tex], [tex]b=2[/tex], and [tex]c=1[/tex] gives [tex]a^3+b^3+c^3=-8+8+1=\boxed1[/tex]

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