Question

The roots of the polynomial f(x)=2x3+20x2+201x+2013 are α,β and γ. What is the value of −(α+1)(β+1)(γ+1)?

Answer 1

By the remainder-factor theorem, we know that f(x)=A(x−α)(x−β)(x−γ). By comparing coefficients, we see that A=2. Substituting x=−1, we get that

1830=2(−1)3+20(−1)2+201(−1)+2013=f(−1)=
2(−1−α)(−1−β)(−1−γ)=−2(α+1)(β+1)(γ+1)
Hence, the value of the expression is equal to 1830/2=915.