Question from pathfinder for Olympiad mathematics 2

Question

If $p$, $q$, $r$ are the real roots of equation $x^3-6x^2+3x+1=0$, determine the possible value of $p^2q+q^2r+r^2p$.

My Attempt:

$p+q+r=6 (1)$

$pq+qr+pr=3 (2)$

$pqr=-1 (3)$

Multiplying (1) and (2) and substituting (3) in it... I came close but still unable to find solution... Please Help

Answer 1

Since $v_1=p^2q+q^2r+r^2p$ is not a symmetric polynomial, there might be multiple values. Another possible value is $v_2=p^2r+q^2p+r^2q.$

Now $v_1+v_2$ and $v_1v_2$ are symmetric. So we can express those values in terms of $p+q+r,$ $pq+pr+qr,$ and $pqr.$

Then solve for the two values, $v_1,v_2.$

I’ll do the harder case. Writing $s_1=p+q+r,$ $s_2=pq+pr+qr,$ and $s_3=pqr,$ we have:

$$\begin{align}v_1v_2=&(p^3+q^3+r^3)s_3+3s_3^2\\&+\left((pq)^3+(qr)^3+(rs)^3\right) \end{align}$$

And $$\begin{align}p^3+q^3+r^3=&s_1^3-3s_1s_2+3s_3\end{align}$$

So that leaves us the last term, $(pq)^3+(pr)^3+(qr)^3.$ One clever way is to write it as:

$$(pq)^3+(pr)^3+(qr)^3=s_3^3\left(\frac1{p^3}+\frac1{q^3}+\frac1{r^3}\right)$$ then reuse the formula for the sum of three cubes we used previously, with $s_1’=s_2/s_3,$ $s_2’=s_1/s_3,$ $s_3’=1/s_3.$ So we have:

$$(pq)^3+(pr)^3+(qr)^3=s_2^3-3s_1s_2s_3+3s_3^2$$

So:

$$\begin{align}v_1v_2&=\left(s_1^3-3s_1s_2+3s_3\right)s_3+3s_3^2+(s_2^3-3s_1s_2s_3+3s_3^2)\\ &=s_1^3s_3-6s_1s_2s_3+s_2^3+9s_3^2\end{align}$$ Since $s_1=6,s_2=3,s_3=-1,$ this means: $$v_1v_2=-6^3+6\cdot 6\cdot 3 +3^3+9=-72.$$

I also get $v_1+v_2=21.$ So:

$$v_i=\frac{21\pm \sqrt{729}}{2}=-3\text{ or }24.$$

A numerical check using the roots provided by Wolfram Alpha indicate this is probably right.

Answer 2

https://qr.ae/pGsCyp This person from Quora has an elegant solution, but did mistake in Vieta's relation (mistake - pqr = 1), and hence arrived to wrong answer. My answer is inspired from this.

Continued from your answer.....

(p+q+r)(pq+qr+pr) = 18

p²q + q²r + r²p + pq² + qr² + rp² + 3pqr = 18

p²q + q²r + r²p = A

pq² + qr² + rp² = B

A + B + 3pqr = 18

A + B = 21

A*B = (p²q + q²r + r²p)(pq² + qr² + rp²)

= (pq)³ + (qr)³ + (rp)³ + 3(pqr)² + pqr(q³ + r³ + p³)

find (pq)³ + (qr)³ + (rp)³ and q³ + r³ + p³ by the algebraic identity a³ + b³ + c³ - 3abc and then (a+b+c)². you will get (pq)³ + (qr)³ + (rp)³ = 84 and q³ + r³ + p³ = 159

A*B = 84 + 3 - 159 = -72

A+B = 21 and A*B = -72

We can form an equation, whose roots are A and B

x² - (A+B)x + (AB) = 0

x² - 21x - 72 = 0

(x + 3) (x - 24) = 0

Put x = A

(A+3)(A-24) = 0

A can be -3 and 24

i.e., the possible values of p²q + q²r + r²p are -3 and 24