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Omni-MATH / The polynomial Q(x)=x³−21x+35 has three different real roots. Find real numbers a and b such that the polynomial x²+ax+b cyclically permutes the roots of Q, that is, if r, s and t …

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problem

The polynomial Q(x)=x3−21x+35Q(x)=x^3-21x+35 has three different real roots. Find real numbers aa and bb such that the polynomial x2+ax+bx^2+ax+b cyclically permutes the roots of QQ, that is, if rr, ss and tt are the roots of QQ (in some order) then P(r)=sP(r)=s, P(s)=tP(s)=t and P(t)=rP(t)=r.
Plain-text mathematical notation (without MathML)
The polynomial Q(x)=x³−21x+35 has three different real roots. Find real numbers a and b such that the polynomial x²+ax+b cyclically permutes the roots of Q, that is, if r, s and t are the roots of Q (in some order) then P(r)=s, P(s)=t and P(t)=r.
Original LaTeX notation
The polynomial $Q(x)=x^3-21x+35$ has three different real roots. Find real numbers $a$ and $b$ such that the polynomial $x^2+ax+b$ cyclically permutes the roots of $Q$, that is, if $r$, $s$ and $t$ are the roots of $Q$ (in some order) then $P(r)=s$, $P(s)=t$ and $P(t)=r$.

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task_id
0ebeb99e-669f-52ee-a028-abf78a9709b6
task_key
test--0ebeb99e-669f-52ee-a028-abf78a9709b6
task_revision_id
1
config / split
/ test

Official source

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