Question

If β and γ are the zeros of the polynomial ax^3 + bx^2 + cx + d, then the value of α^2 + β^2 + γ^2 is _________

(a) \(\frac {b^2-2c}{a^2}\)

(b) \(\frac {b^2-2ca}{a}\)

(c) \(\frac {b^2-2ca}{a^2}\)

(d) \(\frac {b^2+2ca}{a^2}\)

The question was asked in an interview.

My question is based upon Zeros and Coefficients of Polynomial topic in portion Polynomials of Mathematics – Class 10

Answer 1

Right option is (c) \(\frac {b^2-2ca}{a^2}\)

For explanation: β and γ are the zeros of the polynomial ax^3 + bx^2 + cx + d

So, α + β + γ = \(\frac {-b}{a}\)

αβ + βγ + γα = \(\frac {c}{a}\)

Now, α^2 + β^2 + γ^2 = (α + β + γ)^2 – 2(αβ + βγ + γα)

α^2 + β^2 + γ^2 = \((\frac {-b}{a})\)^2 – 2\((\frac {c}{a}) = \frac {b^2}{a^2}\) – 2 \(\frac {c}{a} = \frac {b^2-2ca}{a^2}\)