Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $f(x) = x^3 + ax + b \in Q[x]$ be irreducible. Show that if $\alpha$ is a root of $f$ then $y = \alpha^2$ satisfies $y(y + a)^2 = b^2$ Deduce that $\Delta^2(f) = −(4a^3 + 27b^2)$.

I have found the solutions using other methods. What I would like to know is why "Show that if $\alpha$ is a root of $f$ then $y = \alpha^2$ satisfies $y(y + a)^2 = b^2$" is given and how can solve it using derivatives of $f$ as well as generalize the solutions for other cubic polynomials.

share|improve this question

1 Answer 1

$2y(y+a) = 2y^2 + 2ay = 2\alpha^4 + 2a\alpha^2 = 2\alpha\cdot \alpha^3 + 2a\alpha^2 = 2\alpha(-a\alpha - b) + 2a\alpha^2 = -2b\alpha \to y(y+a) = -b\alpha \to y^2(y+a)^2 = b^2\alpha^2 = yb^2 \to y(y+a)^2 = b^2$.

Note: there is a typo in your post that I changed from $y(y+a)2 = b^2$ to $y(y+a)^2 = b^2$

share|improve this answer
    
but Why it is given? How it is related to discriminant? Thanks for the correction.. –  nnumberr May 27 at 18:24

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.