Mixing
Finding the sum of squares of roots of a quartic polynomial.
Clash Royale CLAN TAG#URR8PPP
up vote
5
down vote
favorite
What is the sum of the squares of the roots of $ x^4 - 8x^3 + 16x^2 - 11x + 5 $ ?
This question is from the 2nd qualifying round of last year's Who Wants to be a Mathematician high school competition which can be seen here:
I know the answer (32) because that is also given in the link, and I have checked by brute force that the given answer is correct.
However, I have made no progress at all in figuring out how to calculate the sum of squares of the roots - either before or after knowing the answer! I was expecting there to be a nice "trick" analagous to the situation if they had given a quadratic and asked the same question -- in that case I know how to get the sum and product of the roots directly from the coefficients, and then a simple bit of algebraic manipulation to arrive at the sum of squares of the roots.
In this case (the quartic) I have no idea how to approach it, and I have not spotted any way to simplify the problem (e.g. I cannot see an obvious factorisation, which might have helped me).
I've looked on the web at various articles which dicuss the relationships between the coefficients of polynomials and their roots and - simply put - I found nothing which gave me inspiration for this puzzle.
Given the audience for this test, it should be susceptible to elementary methods ... I would appreciate any hints and/or solutions!
Thank you.
polynomials
asked 2 hours ago
BBO555
284
New contributor
BBO555 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |Â
up vote
5
down vote
favorite
What is the sum of the squares of the roots of $ x^4 - 8x^3 + 16x^2 - 11x + 5 $ ?
This question is from the 2nd qualifying round of last year's Who Wants to be a Mathematician high school competition which can be seen here:
I know the answer (32) because that is also given in the link, and I have checked by brute force that the given answer is correct.
However, I have made no progress at all in figuring out how to calculate the sum of squares of the roots - either before or after knowing the answer! I was expecting there to be a nice "trick" analagous to the situation if they had given a quadratic and asked the same question -- in that case I know how to get the sum and product of the roots directly from the coefficients, and then a simple bit of algebraic manipulation to arrive at the sum of squares of the roots.
In this case (the quartic) I have no idea how to approach it, and I have not spotted any way to simplify the problem (e.g. I cannot see an obvious factorisation, which might have helped me).
I've looked on the web at various articles which dicuss the relationships between the coefficients of polynomials and their roots and - simply put - I found nothing which gave me inspiration for this puzzle.
Given the audience for this test, it should be susceptible to elementary methods ... I would appreciate any hints and/or solutions!
Thank you.
polynomials
asked 2 hours ago
BBO555
284
New contributor
BBO555 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$(sum a_i)^2 = sum a_i^2 + sum_limitsineq j a_ia_j$
– AnotherJohnDoe
1 hour ago
This is a wonderful post, a rare essay for a new contributor. Since finding the solution is in this case more important than having it, here is a hint: Use the relations of Vieta for the four roots $a,b,c,d$ - say - of the given equation, which show how to extract the elementary symmetric polynomials, first two being $sum a =a+b+c+d$ and $sum ab=ab+ac+ad+bc+bd+cd$. Can you now find $(a+b+c+d)^2$? en.wikipedia.org/wiki/Vieta%27s_formulas
– dan_fulea
1 hour ago
Thank you! This, and the answers posted soon afterwards were very helpful.
– BBO555
1 hour ago
add a comment |Â
up vote
5
down vote
favorite
up vote
5
down vote
favorite
What is the sum of the squares of the roots of $ x^4 - 8x^3 + 16x^2 - 11x + 5 $ ?
This question is from the 2nd qualifying round of last year's Who Wants to be a Mathematician high school competition which can be seen here:
I know the answer (32) because that is also given in the link, and I have checked by brute force that the given answer is correct.
However, I have made no progress at all in figuring out how to calculate the sum of squares of the roots - either before or after knowing the answer! I was expecting there to be a nice "trick" analagous to the situation if they had given a quadratic and asked the same question -- in that case I know how to get the sum and product of the roots directly from the coefficients, and then a simple bit of algebraic manipulation to arrive at the sum of squares of the roots.
In this case (the quartic) I have no idea how to approach it, and I have not spotted any way to simplify the problem (e.g. I cannot see an obvious factorisation, which might have helped me).
I've looked on the web at various articles which dicuss the relationships between the coefficients of polynomials and their roots and - simply put - I found nothing which gave me inspiration for this puzzle.
Given the audience for this test, it should be susceptible to elementary methods ... I would appreciate any hints and/or solutions!
Thank you.
polynomials
asked 2 hours ago
BBO555
284
New contributor
BBO555 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
What is the sum of the squares of the roots of $ x^4 - 8x^3 + 16x^2 - 11x + 5 $ ?
This question is from the 2nd qualifying round of last year's Who Wants to be a Mathematician high school competition which can be seen here:
I know the answer (32) because that is also given in the link, and I have checked by brute force that the given answer is correct.
However, I have made no progress at all in figuring out how to calculate the sum of squares of the roots - either before or after knowing the answer! I was expecting there to be a nice "trick" analagous to the situation if they had given a quadratic and asked the same question -- in that case I know how to get the sum and product of the roots directly from the coefficients, and then a simple bit of algebraic manipulation to arrive at the sum of squares of the roots.
In this case (the quartic) I have no idea how to approach it, and I have not spotted any way to simplify the problem (e.g. I cannot see an obvious factorisation, which might have helped me).
I've looked on the web at various articles which dicuss the relationships between the coefficients of polynomials and their roots and - simply put - I found nothing which gave me inspiration for this puzzle.
Given the audience for this test, it should be susceptible to elementary methods ... I would appreciate any hints and/or solutions!
Thank you.
polynomials
polynomials
asked 2 hours ago
BBO555
284
New contributor
BBO555 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 hours ago
BBO555
284
New contributor
BBO555 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 hours ago
BBO555
284
New contributor
BBO555 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 hours ago
BBO555
284
asked 2 hours ago
BBO555
284
284
New contributor
BBO555 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
BBO555 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
BBO555 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$(sum a_i)^2 = sum a_i^2 + sum_limitsineq j a_ia_j$
– AnotherJohnDoe
1 hour ago
This is a wonderful post, a rare essay for a new contributor. Since finding the solution is in this case more important than having it, here is a hint: Use the relations of Vieta for the four roots $a,b,c,d$ - say - of the given equation, which show how to extract the elementary symmetric polynomials, first two being $sum a =a+b+c+d$ and $sum ab=ab+ac+ad+bc+bd+cd$. Can you now find $(a+b+c+d)^2$? en.wikipedia.org/wiki/Vieta%27s_formulas
– dan_fulea
1 hour ago
Thank you! This, and the answers posted soon afterwards were very helpful.
– BBO555
1 hour ago
add a comment |Â
$(sum a_i)^2 = sum a_i^2 + sum_limitsineq j a_ia_j$
– AnotherJohnDoe
1 hour ago
This is a wonderful post, a rare essay for a new contributor. Since finding the solution is in this case more important than having it, here is a hint: Use the relations of Vieta for the four roots $a,b,c,d$ - say - of the given equation, which show how to extract the elementary symmetric polynomials, first two being $sum a =a+b+c+d$ and $sum ab=ab+ac+ad+bc+bd+cd$. Can you now find $(a+b+c+d)^2$? en.wikipedia.org/wiki/Vieta%27s_formulas
– dan_fulea
1 hour ago
Thank you! This, and the answers posted soon afterwards were very helpful.
– BBO555
1 hour ago
$(sum a_i)^2 = sum a_i^2 + sum_limitsineq j a_ia_j$
– AnotherJohnDoe
1 hour ago
$(sum a_i)^2 = sum a_i^2 + sum_limitsineq j a_ia_j$
– AnotherJohnDoe
1 hour ago
This is a wonderful post, a rare essay for a new contributor. Since finding the solution is in this case more important than having it, here is a hint: Use the relations of Vieta for the four roots $a,b,c,d$ - say - of the given equation, which show how to extract the elementary symmetric polynomials, first two being $sum a =a+b+c+d$ and $sum ab=ab+ac+ad+bc+bd+cd$. Can you now find $(a+b+c+d)^2$? en.wikipedia.org/wiki/Vieta%27s_formulas
– dan_fulea
1 hour ago
This is a wonderful post, a rare essay for a new contributor. Since finding the solution is in this case more important than having it, here is a hint: Use the relations of Vieta for the four roots $a,b,c,d$ - say - of the given equation, which show how to extract the elementary symmetric polynomials, first two being $sum a =a+b+c+d$ and $sum ab=ab+ac+ad+bc+bd+cd$. Can you now find $(a+b+c+d)^2$? en.wikipedia.org/wiki/Vieta%27s_formulas
– dan_fulea
1 hour ago
Thank you! This, and the answers posted soon afterwards were very helpful.
– BBO555
1 hour ago
Thank you! This, and the answers posted soon afterwards were very helpful.
– BBO555
1 hour ago
add a comment |Â
3 Answers
3
active
oldest
votes
up vote
4
down vote
accepted
We have that
$$(x-a)(x-b)(x-c)(x-d)=$$
$$=x^4-(a+b+c+d)x^3+(ab+ac+ad+bc+bd+cd)x^2\-(abc+abd+acd+bcd)x+abcd$$
then by
- $S_1=a+b+c+d=0$
- $S_2=ab+ac+ad+bc+bd+cd=0$
- $S_3=abc+abd+acd+bcd=-1$
- $S_4=abcd=1$
$$a^2+b^2+c^2+d^2=S_1^2-2S_2$$
and more in general by Newton's sums we have that
- $P_1=a+b+c+d=S_1=0$
- $P_2=a^2+b^2+c^2+d^2=S_1P_1-2S_2=0$
- $P_3=a^3+b^3+c^3+d^3=S_1P_2-S_3P_1+3S_3=-3$
- $P_4=a^4+b^4+c^4+d^4=S_1P_3-S_2P_2+S_3P_1-4S_4=-4$
answered 1 hour ago
gimusi
72.7k73888
Thanks for this very quick and helpful answer !
– BBO555
1 hour ago
You are welcome! Bye
– gimusi
55 mins ago
add a comment |Â
up vote
3
down vote
Hint:
$$r_0^2+r_1^2+r_2^2+r_3^2=(r_0+r_1+r_2+r_3)^2-2(r_0r_1+r_0r_2+r_0r_3+r_1r_2+r_1r_3+r_2r_3)$$
and Vieta.
answered 1 hour ago
Yves Daoust
114k666209
Thank you! It appears the solution was much closer to hand than I realised at first ...
– BBO555
1 hour ago
add a comment |Â
up vote
2
down vote
Hint:
beginalign
sum_i=1^4a_i^2 &= left(sum_i=1^4a_i right)^2-2sum_i< ja_ia_j
endalign
Also Vieta's formula might help.
answered 1 hour ago
Siong Thye Goh
82.3k1456104
Yes indeed - thank you!
– BBO555
1 hour ago
add a comment |Â
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
accepted
We have that
$$(x-a)(x-b)(x-c)(x-d)=$$
$$=x^4-(a+b+c+d)x^3+(ab+ac+ad+bc+bd+cd)x^2\-(abc+abd+acd+bcd)x+abcd$$
then by
- $S_1=a+b+c+d=0$
- $S_2=ab+ac+ad+bc+bd+cd=0$
- $S_3=abc+abd+acd+bcd=-1$
- $S_4=abcd=1$
$$a^2+b^2+c^2+d^2=S_1^2-2S_2$$
and more in general by Newton's sums we have that
- $P_1=a+b+c+d=S_1=0$
- $P_2=a^2+b^2+c^2+d^2=S_1P_1-2S_2=0$
- $P_3=a^3+b^3+c^3+d^3=S_1P_2-S_3P_1+3S_3=-3$
- $P_4=a^4+b^4+c^4+d^4=S_1P_3-S_2P_2+S_3P_1-4S_4=-4$
answered 1 hour ago
gimusi
72.7k73888
Thanks for this very quick and helpful answer !
– BBO555
1 hour ago
You are welcome! Bye
– gimusi
55 mins ago
add a comment |Â
up vote
4
down vote
accepted
We have that
$$(x-a)(x-b)(x-c)(x-d)=$$
$$=x^4-(a+b+c+d)x^3+(ab+ac+ad+bc+bd+cd)x^2\-(abc+abd+acd+bcd)x+abcd$$
then by
- $S_1=a+b+c+d=0$
- $S_2=ab+ac+ad+bc+bd+cd=0$
- $S_3=abc+abd+acd+bcd=-1$
- $S_4=abcd=1$
$$a^2+b^2+c^2+d^2=S_1^2-2S_2$$
and more in general by Newton's sums we have that
- $P_1=a+b+c+d=S_1=0$
- $P_2=a^2+b^2+c^2+d^2=S_1P_1-2S_2=0$
- $P_3=a^3+b^3+c^3+d^3=S_1P_2-S_3P_1+3S_3=-3$
- $P_4=a^4+b^4+c^4+d^4=S_1P_3-S_2P_2+S_3P_1-4S_4=-4$
answered 1 hour ago
gimusi
72.7k73888
Thanks for this very quick and helpful answer !
– BBO555
1 hour ago
You are welcome! Bye
– gimusi
55 mins ago
add a comment |Â
up vote
4
down vote
accepted
up vote
4
down vote
accepted
We have that
$$(x-a)(x-b)(x-c)(x-d)=$$
$$=x^4-(a+b+c+d)x^3+(ab+ac+ad+bc+bd+cd)x^2\-(abc+abd+acd+bcd)x+abcd$$
then by
- $S_1=a+b+c+d=0$
- $S_2=ab+ac+ad+bc+bd+cd=0$
- $S_3=abc+abd+acd+bcd=-1$
- $S_4=abcd=1$
$$a^2+b^2+c^2+d^2=S_1^2-2S_2$$
and more in general by Newton's sums we have that
- $P_1=a+b+c+d=S_1=0$
- $P_2=a^2+b^2+c^2+d^2=S_1P_1-2S_2=0$
- $P_3=a^3+b^3+c^3+d^3=S_1P_2-S_3P_1+3S_3=-3$
- $P_4=a^4+b^4+c^4+d^4=S_1P_3-S_2P_2+S_3P_1-4S_4=-4$
answered 1 hour ago
gimusi
72.7k73888
We have that
$$(x-a)(x-b)(x-c)(x-d)=$$
$$=x^4-(a+b+c+d)x^3+(ab+ac+ad+bc+bd+cd)x^2\-(abc+abd+acd+bcd)x+abcd$$
then by
- $S_1=a+b+c+d=0$
- $S_2=ab+ac+ad+bc+bd+cd=0$
- $S_3=abc+abd+acd+bcd=-1$
- $S_4=abcd=1$
$$a^2+b^2+c^2+d^2=S_1^2-2S_2$$
and more in general by Newton's sums we have that
- $P_1=a+b+c+d=S_1=0$
- $P_2=a^2+b^2+c^2+d^2=S_1P_1-2S_2=0$
- $P_3=a^3+b^3+c^3+d^3=S_1P_2-S_3P_1+3S_3=-3$
- $P_4=a^4+b^4+c^4+d^4=S_1P_3-S_2P_2+S_3P_1-4S_4=-4$
answered 1 hour ago
gimusi
72.7k73888
edited 55 mins ago
answered 1 hour ago
gimusi
72.7k73888
answered 1 hour ago
gimusi
72.7k73888
answered 1 hour ago
gimusi
72.7k73888
72.7k73888
Thanks for this very quick and helpful answer !
– BBO555
1 hour ago
You are welcome! Bye
– gimusi
55 mins ago
add a comment |Â
Thanks for this very quick and helpful answer !
– BBO555
1 hour ago
You are welcome! Bye
– gimusi
55 mins ago
Thanks for this very quick and helpful answer !
– BBO555
1 hour ago
Thanks for this very quick and helpful answer !
– BBO555
1 hour ago
You are welcome! Bye
– gimusi
55 mins ago
You are welcome! Bye
– gimusi
55 mins ago
add a comment |Â
up vote
3
down vote
Hint:
$$r_0^2+r_1^2+r_2^2+r_3^2=(r_0+r_1+r_2+r_3)^2-2(r_0r_1+r_0r_2+r_0r_3+r_1r_2+r_1r_3+r_2r_3)$$
and Vieta.
answered 1 hour ago
Yves Daoust
114k666209
Thank you! It appears the solution was much closer to hand than I realised at first ...
– BBO555
1 hour ago
add a comment |Â
up vote
3
down vote
Hint:
$$r_0^2+r_1^2+r_2^2+r_3^2=(r_0+r_1+r_2+r_3)^2-2(r_0r_1+r_0r_2+r_0r_3+r_1r_2+r_1r_3+r_2r_3)$$
and Vieta.
answered 1 hour ago
Yves Daoust
114k666209
Thank you! It appears the solution was much closer to hand than I realised at first ...
– BBO555
1 hour ago
add a comment |Â
up vote
3
down vote
up vote
3
down vote
Hint:
$$r_0^2+r_1^2+r_2^2+r_3^2=(r_0+r_1+r_2+r_3)^2-2(r_0r_1+r_0r_2+r_0r_3+r_1r_2+r_1r_3+r_2r_3)$$
and Vieta.
answered 1 hour ago
Yves Daoust
114k666209
Hint:
$$r_0^2+r_1^2+r_2^2+r_3^2=(r_0+r_1+r_2+r_3)^2-2(r_0r_1+r_0r_2+r_0r_3+r_1r_2+r_1r_3+r_2r_3)$$
and Vieta.
answered 1 hour ago
Yves Daoust
114k666209
answered 1 hour ago
Yves Daoust
114k666209
answered 1 hour ago
Yves Daoust
114k666209
answered 1 hour ago
Yves Daoust
114k666209
114k666209
Thank you! It appears the solution was much closer to hand than I realised at first ...
– BBO555
1 hour ago
add a comment |Â
Thank you! It appears the solution was much closer to hand than I realised at first ...
– BBO555
1 hour ago
Thank you! It appears the solution was much closer to hand than I realised at first ...
– BBO555
1 hour ago
Thank you! It appears the solution was much closer to hand than I realised at first ...
– BBO555
1 hour ago
add a comment |Â
up vote
2
down vote
Hint:
beginalign
sum_i=1^4a_i^2 &= left(sum_i=1^4a_i right)^2-2sum_i< ja_ia_j
endalign
Also Vieta's formula might help.
answered 1 hour ago
Siong Thye Goh
82.3k1456104
Yes indeed - thank you!
– BBO555
1 hour ago
add a comment |Â
up vote
2
down vote
Hint:
beginalign
sum_i=1^4a_i^2 &= left(sum_i=1^4a_i right)^2-2sum_i< ja_ia_j
endalign
Also Vieta's formula might help.
answered 1 hour ago
Siong Thye Goh
82.3k1456104
Yes indeed - thank you!
– BBO555
1 hour ago
add a comment |Â
up vote
2
down vote
up vote
2
down vote
Hint:
beginalign
sum_i=1^4a_i^2 &= left(sum_i=1^4a_i right)^2-2sum_i< ja_ia_j
endalign
Also Vieta's formula might help.
answered 1 hour ago
Siong Thye Goh
82.3k1456104
Hint:
beginalign
sum_i=1^4a_i^2 &= left(sum_i=1^4a_i right)^2-2sum_i< ja_ia_j
endalign
Also Vieta's formula might help.
answered 1 hour ago
Siong Thye Goh
82.3k1456104
answered 1 hour ago
Siong Thye Goh
82.3k1456104
answered 1 hour ago
Siong Thye Goh
82.3k1456104
answered 1 hour ago
Siong Thye Goh
82.3k1456104
82.3k1456104
Yes indeed - thank you!
– BBO555
1 hour ago
add a comment |Â
Yes indeed - thank you!
– BBO555
1 hour ago
Yes indeed - thank you!
– BBO555
1 hour ago
Yes indeed - thank you!
– BBO555
1 hour ago
add a comment |Â
BBO555 is a new contributor. Be nice, and check out our Code of Conduct.
BBO555 is a new contributor. Be nice, and check out our Code of Conduct.
BBO555 is a new contributor. Be nice, and check out our Code of Conduct.
BBO555 is a new contributor. Be nice, and check out our Code of Conduct.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2920175%2ffinding-the-sum-of-squares-of-roots-of-a-quartic-polynomial%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
$(sum a_i)^2 = sum a_i^2 + sum_limitsineq j a_ia_j$
– AnotherJohnDoe
1 hour ago
This is a wonderful post, a rare essay for a new contributor. Since finding the solution is in this case more important than having it, here is a hint: Use the relations of Vieta for the four roots $a,b,c,d$ - say - of the given equation, which show how to extract the elementary symmetric polynomials, first two being $sum a =a+b+c+d$ and $sum ab=ab+ac+ad+bc+bd+cd$. Can you now find $(a+b+c+d)^2$? en.wikipedia.org/wiki/Vieta%27s_formulas
– dan_fulea
1 hour ago
Thank you! This, and the answers posted soon afterwards were very helpful.
– BBO555
1 hour ago