When do roots of three quadratic polynomials multiply

Tianna Costa

Tianna Costa

Answered question

2022-03-14

When do roots of three quadratic polynomials multiply to 1?
Say I have a trio of quadratic polynomials p1,p2,p3. Under what conditions will I have r1r2r3=1 where each pi(ri)=0
p1(r1)=0, p2(r2)=0, p3(r3)=0, r1r2r3=1

Answer & Explanation

conduchafr4

conduchafr4

Beginner2022-03-15Added 5 answers

Step 1
Your question asks if you have three quadratic polynomials
p1(x)!=!a1x2+b1x+c1,;;p2(x)!=!a2x2+b2x+c2,;;p3(x)!=!a3x3+b3x+c3
with three pairs of roots
p1(r1+)=p1(r1)=0,p2(r2+)=p2(r2)=0,p3(r3+)=p3(r3)=0
where for n=1,2,3,
rn,±=bn±bn24ancn2an,
Step 2
then what is the condition that ,r1r2r3=1, for some choice of the roots as given in terms of the coefficients of the three polynomials? The answer is given by a homogeneous degree 12 polynomial expanded out with 34 monomial terms
P=(a1a2a3)4i,j,k=±(1r1,i,r2,j,r3,k)=(a1a2a3)4++(c1c2c3)4
where the … represents the other 32 degree-12 monomial terms. I used a computer algebra system to get the expansion. As stated in the question and I don't think it can be simplified except for special cases, but I have been wrong before, so maybe there is hope.

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