I'm not quite too sure how to approach this question, so any explanation using any technique would be greatly appreciated. Thank you!
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Graph! This is the simplest technique if you are asking for any technique. – Yash Jain Feb 06 '18 at 03:18
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3If I want to understand the technique behind approaching these questions, how would I approach it algebraically then? – Shizukesa Feb 06 '18 at 03:20
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Hint: it cannot have $\,4\,$ distinct real roots. Either use Descartes' rule of signs, or look at $\,P''(x)\,$.
dxiv
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Assume it has 4 real roots $a,b,c,d$. By vieta's formulas,
$$abcd=m^2$$ $$a+b+c+d=0$$ $$ab+bc+cd+da+ac+bd=0$$ $$abc+bcd+cda+dab=m-3$$
But since $(a+b+c+d)^2=a^2+b^2+c^2+d^2+2(ab+bc+cd+da+ac+bd)$,
$$a^2+b^2+c^2+d^2=0\implies a=b=c=d=0$$
So this means if $P(x)$ has all real roots they must be all $0$, contradicting distinctness.
Therefore no value of $m$ makes $P(x)$ have 4 distinct real roots.
Akababa
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