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

Question

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

Answer 1

Consider the cubic polynomial,

`(x+a)(x+b)(x+c)`

Expanding this gives

`x^3+(a+b+c)x^2+(ab+ac+bc)x+abc=x^3+x^2-4x-4`

We can factor this by grouping,

`x^3+x^2-4x-4=x^2(x+1)-4(x+1)=(x^2-4)(x+1)=(x-2)(x+2)(x+1)`

Then letting
`a=-2`,
`b=2`, and
`c=1` gives
`a^3+b^3+c^3=-8+8+1=\boxed1`