Calculate $sum^20_k=1frac1x_k-x_k^2$ where $x_k$ are roots of $P(x)=x^20+x^10+x^5+2$Sum of cubed rootsProblem: Sum of absolute values of polynomial rootsSum of squares of roots of a polynomial $P(x)$Polynomial with real rootsHomework: Sum of the cubed roots of polynomialSum of roots rational but product irrationalPolynomial problemsCalculate sum of roots$ fracQleft(x_1right)P'left(x_1right)+fracQleft(x_2right)P'left(x_2right)+fracQleft(x_3right)P'left(x_3right)=? $If roots of $x^6=p(x)$ are given then choose the correct option
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Calculate $sum^20_k=1frac1x_k-x_k^2$ where $x_k$ are roots of $P(x)=x^20+x^10+x^5+2$
Sum of cubed rootsProblem: Sum of absolute values of polynomial rootsSum of squares of roots of a polynomial $P(x)$Polynomial with real rootsHomework: Sum of the cubed roots of polynomialSum of roots rational but product irrationalPolynomial problemsCalculate sum of roots$ fracQleft(x_1right)P'left(x_1right)+fracQleft(x_2right)P'left(x_2right)+fracQleft(x_3right)P'left(x_3right)=? $If roots of $x^6=p(x)$ are given then choose the correct option
$begingroup$
We have the polynomial $P(x)=x^20+x^10+x^5+2$, which has roots $x_1,x_2,x_3,...,x_20$. Calculate the sum $$sum^20_k=1frac1x_k-x_k^2$$
What I've noticed: $$sum^20_k=1frac1x_k-x_k^2=sum^20_k=1left(frac1x_k+frac11-x_kright)$$
I know how to calculate the first sum: $sum^20_k=1frac1x_k$.
Please help me calculate the second one: $sum^20_k=1frac11-x_k$.
polynomials contest-math roots
edited Mar 23 at 16:13
Jean Marie
32k42355
asked Mar 22 at 19:22
P. MillerP. Miller
462
$endgroup$
add a comment |
$begingroup$
We have the polynomial $P(x)=x^20+x^10+x^5+2$, which has roots $x_1,x_2,x_3,...,x_20$. Calculate the sum $$sum^20_k=1frac1x_k-x_k^2$$
What I've noticed: $$sum^20_k=1frac1x_k-x_k^2=sum^20_k=1left(frac1x_k+frac11-x_kright)$$
I know how to calculate the first sum: $sum^20_k=1frac1x_k$.
Please help me calculate the second one: $sum^20_k=1frac11-x_k$.
polynomials contest-math roots
edited Mar 23 at 16:13
Jean Marie
32k42355
asked Mar 22 at 19:22
P. MillerP. Miller
462
$endgroup$
2
$begingroup$
If $x_k$ are roots of $P(x)$, then $y_k=1-x_k$ are roots of $P(1-x)$. Maybe that can help?
$endgroup$
– Sil
Mar 22 at 19:38
6
$begingroup$
Hint: $fracP'(x)P(x) = sum_k=1^20frac1x-x_k$
$endgroup$
– achille hui
Mar 22 at 19:44
$begingroup$
Some of your tags have no connection with the issue : linear-algebra, symmetric polynomials (we don't have a symmetry of coefficients), abstract algebra (no group, ring, field, etc... here). I have taken the liberty to suppress them. I have added the "roots" tags.
$endgroup$
– Jean Marie
Mar 23 at 16:11
add a comment |
$begingroup$
We have the polynomial $P(x)=x^20+x^10+x^5+2$, which has roots $x_1,x_2,x_3,...,x_20$. Calculate the sum $$sum^20_k=1frac1x_k-x_k^2$$
What I've noticed: $$sum^20_k=1frac1x_k-x_k^2=sum^20_k=1left(frac1x_k+frac11-x_kright)$$
I know how to calculate the first sum: $sum^20_k=1frac1x_k$.
Please help me calculate the second one: $sum^20_k=1frac11-x_k$.
polynomials contest-math roots
edited Mar 23 at 16:13
Jean Marie
32k42355
asked Mar 22 at 19:22
P. MillerP. Miller
462
$endgroup$
We have the polynomial $P(x)=x^20+x^10+x^5+2$, which has roots $x_1,x_2,x_3,...,x_20$. Calculate the sum $$sum^20_k=1frac1x_k-x_k^2$$
What I've noticed: $$sum^20_k=1frac1x_k-x_k^2=sum^20_k=1left(frac1x_k+frac11-x_kright)$$
I know how to calculate the first sum: $sum^20_k=1frac1x_k$.
Please help me calculate the second one: $sum^20_k=1frac11-x_k$.
polynomials contest-math roots
polynomials contest-math roots
edited Mar 23 at 16:13
Jean Marie
32k42355
asked Mar 22 at 19:22
P. MillerP. Miller
462
edited Mar 23 at 16:13
Jean Marie
32k42355
asked Mar 22 at 19:22
P. MillerP. Miller
462
edited Mar 23 at 16:13
Jean Marie
32k42355
edited Mar 23 at 16:13
Jean Marie
32k42355
edited Mar 23 at 16:13
Jean Marie
32k42355
32k42355
asked Mar 22 at 19:22
P. MillerP. Miller
462
asked Mar 22 at 19:22
P. MillerP. Miller
462
asked Mar 22 at 19:22
P. MillerP. Miller
462
462
2
$begingroup$
If $x_k$ are roots of $P(x)$, then $y_k=1-x_k$ are roots of $P(1-x)$. Maybe that can help?
$endgroup$
– Sil
Mar 22 at 19:38
6
$begingroup$
Hint: $fracP'(x)P(x) = sum_k=1^20frac1x-x_k$
$endgroup$
– achille hui
Mar 22 at 19:44
$begingroup$
Some of your tags have no connection with the issue : linear-algebra, symmetric polynomials (we don't have a symmetry of coefficients), abstract algebra (no group, ring, field, etc... here). I have taken the liberty to suppress them. I have added the "roots" tags.
$endgroup$
– Jean Marie
Mar 23 at 16:11
add a comment |
2
$begingroup$
If $x_k$ are roots of $P(x)$, then $y_k=1-x_k$ are roots of $P(1-x)$. Maybe that can help?
$endgroup$
– Sil
Mar 22 at 19:38
6
$begingroup$
Hint: $fracP'(x)P(x) = sum_k=1^20frac1x-x_k$
$endgroup$
– achille hui
Mar 22 at 19:44
$begingroup$
Some of your tags have no connection with the issue : linear-algebra, symmetric polynomials (we don't have a symmetry of coefficients), abstract algebra (no group, ring, field, etc... here). I have taken the liberty to suppress them. I have added the "roots" tags.
$endgroup$
– Jean Marie
Mar 23 at 16:11
2
2
$begingroup$
If $x_k$ are roots of $P(x)$, then $y_k=1-x_k$ are roots of $P(1-x)$. Maybe that can help?
$endgroup$
– Sil
Mar 22 at 19:38
$begingroup$
If $x_k$ are roots of $P(x)$, then $y_k=1-x_k$ are roots of $P(1-x)$. Maybe that can help?
$endgroup$
– Sil
Mar 22 at 19:38
6
6
$begingroup$
Hint: $fracP'(x)P(x) = sum_k=1^20frac1x-x_k$
$endgroup$
– achille hui
Mar 22 at 19:44
$begingroup$
Hint: $fracP'(x)P(x) = sum_k=1^20frac1x-x_k$
$endgroup$
– achille hui
Mar 22 at 19:44
$begingroup$
Some of your tags have no connection with the issue : linear-algebra, symmetric polynomials (we don't have a symmetry of coefficients), abstract algebra (no group, ring, field, etc... here). I have taken the liberty to suppress them. I have added the "roots" tags.
$endgroup$
– Jean Marie
Mar 23 at 16:11
$begingroup$
Some of your tags have no connection with the issue : linear-algebra, symmetric polynomials (we don't have a symmetry of coefficients), abstract algebra (no group, ring, field, etc... here). I have taken the liberty to suppress them. I have added the "roots" tags.
$endgroup$
– Jean Marie
Mar 23 at 16:11
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Since $$fracP'(x)P(x) = sum_k=1^20frac1x-x_k$$
and $P'(x)= 20x^19+10x^9+5x^4$
we have $$sum_k=1^20frac11-x_k=fracP'(1)P(1) = 35over 5=7$$
answered Mar 22 at 19:50
Maria MazurMaria Mazur
51k1362127
$endgroup$
$begingroup$
Why is $$fracP'(x)P(x) = sum_k=1^20frac1x-x_k$$
$endgroup$
– Dr. Mathva
Mar 24 at 9:34
$begingroup$
Write $P(x)= a_n(x-x_1)(x-x_2)...$ take it log and then derviate it@Dr.Mathva
$endgroup$
– Maria Mazur
Mar 24 at 11:02
add a comment |
$begingroup$
Hint:
Set $y=1-x$. If the $x_k$ satisfy the equation $;x^20+x^10+x^5+2=0$, the corresponding $:y_k$ satisfy the equation
$$(1-y)^20+(1-y)^10+(1-y)^5+2=0.$$
Can you find the constant term and the coefficient of $y$ in this equation, to use Vieta's relations?
answered Mar 22 at 19:44
BernardBernard
125k743119
$endgroup$
add a comment |
Your Answer
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2 Answers
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2 Answers
2
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$begingroup$
Since $$fracP'(x)P(x) = sum_k=1^20frac1x-x_k$$
and $P'(x)= 20x^19+10x^9+5x^4$
we have $$sum_k=1^20frac11-x_k=fracP'(1)P(1) = 35over 5=7$$
answered Mar 22 at 19:50
Maria MazurMaria Mazur
51k1362127
$endgroup$
$begingroup$
Why is $$fracP'(x)P(x) = sum_k=1^20frac1x-x_k$$
$endgroup$
– Dr. Mathva
Mar 24 at 9:34
$begingroup$
Write $P(x)= a_n(x-x_1)(x-x_2)...$ take it log and then derviate it@Dr.Mathva
$endgroup$
– Maria Mazur
Mar 24 at 11:02
add a comment |
$begingroup$
Since $$fracP'(x)P(x) = sum_k=1^20frac1x-x_k$$
and $P'(x)= 20x^19+10x^9+5x^4$
we have $$sum_k=1^20frac11-x_k=fracP'(1)P(1) = 35over 5=7$$
answered Mar 22 at 19:50
Maria MazurMaria Mazur
51k1362127
$endgroup$
$begingroup$
Why is $$fracP'(x)P(x) = sum_k=1^20frac1x-x_k$$
$endgroup$
– Dr. Mathva
Mar 24 at 9:34
$begingroup$
Write $P(x)= a_n(x-x_1)(x-x_2)...$ take it log and then derviate it@Dr.Mathva
$endgroup$
– Maria Mazur
Mar 24 at 11:02
add a comment |
$begingroup$
Since $$fracP'(x)P(x) = sum_k=1^20frac1x-x_k$$
and $P'(x)= 20x^19+10x^9+5x^4$
we have $$sum_k=1^20frac11-x_k=fracP'(1)P(1) = 35over 5=7$$
answered Mar 22 at 19:50
Maria MazurMaria Mazur
51k1362127
$endgroup$
Since $$fracP'(x)P(x) = sum_k=1^20frac1x-x_k$$
and $P'(x)= 20x^19+10x^9+5x^4$
we have $$sum_k=1^20frac11-x_k=fracP'(1)P(1) = 35over 5=7$$
answered Mar 22 at 19:50
Maria MazurMaria Mazur
51k1362127
answered Mar 22 at 19:50
Maria MazurMaria Mazur
51k1362127
answered Mar 22 at 19:50
Maria MazurMaria Mazur
51k1362127
answered Mar 22 at 19:50
Maria MazurMaria Mazur
51k1362127
51k1362127
$begingroup$
Why is $$fracP'(x)P(x) = sum_k=1^20frac1x-x_k$$
$endgroup$
– Dr. Mathva
Mar 24 at 9:34
$begingroup$
Write $P(x)= a_n(x-x_1)(x-x_2)...$ take it log and then derviate it@Dr.Mathva
$endgroup$
– Maria Mazur
Mar 24 at 11:02
add a comment |
$begingroup$
Why is $$fracP'(x)P(x) = sum_k=1^20frac1x-x_k$$
$endgroup$
– Dr. Mathva
Mar 24 at 9:34
$begingroup$
Write $P(x)= a_n(x-x_1)(x-x_2)...$ take it log and then derviate it@Dr.Mathva
$endgroup$
– Maria Mazur
Mar 24 at 11:02
$begingroup$
Why is $$fracP'(x)P(x) = sum_k=1^20frac1x-x_k$$
$endgroup$
– Dr. Mathva
Mar 24 at 9:34
$begingroup$
Why is $$fracP'(x)P(x) = sum_k=1^20frac1x-x_k$$
$endgroup$
– Dr. Mathva
Mar 24 at 9:34
$begingroup$
Write $P(x)= a_n(x-x_1)(x-x_2)...$ take it log and then derviate it@Dr.Mathva
$endgroup$
– Maria Mazur
Mar 24 at 11:02
$begingroup$
Write $P(x)= a_n(x-x_1)(x-x_2)...$ take it log and then derviate it@Dr.Mathva
$endgroup$
– Maria Mazur
Mar 24 at 11:02
add a comment |
$begingroup$
Hint:
Set $y=1-x$. If the $x_k$ satisfy the equation $;x^20+x^10+x^5+2=0$, the corresponding $:y_k$ satisfy the equation
$$(1-y)^20+(1-y)^10+(1-y)^5+2=0.$$
Can you find the constant term and the coefficient of $y$ in this equation, to use Vieta's relations?
answered Mar 22 at 19:44
BernardBernard
125k743119
$endgroup$
add a comment |
$begingroup$
Hint:
Set $y=1-x$. If the $x_k$ satisfy the equation $;x^20+x^10+x^5+2=0$, the corresponding $:y_k$ satisfy the equation
$$(1-y)^20+(1-y)^10+(1-y)^5+2=0.$$
Can you find the constant term and the coefficient of $y$ in this equation, to use Vieta's relations?
answered Mar 22 at 19:44
BernardBernard
125k743119
$endgroup$
add a comment |
$begingroup$
Hint:
Set $y=1-x$. If the $x_k$ satisfy the equation $;x^20+x^10+x^5+2=0$, the corresponding $:y_k$ satisfy the equation
$$(1-y)^20+(1-y)^10+(1-y)^5+2=0.$$
Can you find the constant term and the coefficient of $y$ in this equation, to use Vieta's relations?
answered Mar 22 at 19:44
BernardBernard
125k743119
$endgroup$
Hint:
Set $y=1-x$. If the $x_k$ satisfy the equation $;x^20+x^10+x^5+2=0$, the corresponding $:y_k$ satisfy the equation
$$(1-y)^20+(1-y)^10+(1-y)^5+2=0.$$
Can you find the constant term and the coefficient of $y$ in this equation, to use Vieta's relations?
answered Mar 22 at 19:44
BernardBernard
125k743119
answered Mar 22 at 19:44
BernardBernard
125k743119
answered Mar 22 at 19:44
BernardBernard
125k743119
answered Mar 22 at 19:44
BernardBernard
125k743119
125k743119
add a comment |
add a comment |
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$begingroup$
If $x_k$ are roots of $P(x)$, then $y_k=1-x_k$ are roots of $P(1-x)$. Maybe that can help?
$endgroup$
– Sil
Mar 22 at 19:38
6
$begingroup$
Hint: $fracP'(x)P(x) = sum_k=1^20frac1x-x_k$
$endgroup$
– achille hui
Mar 22 at 19:44
$begingroup$
Some of your tags have no connection with the issue : linear-algebra, symmetric polynomials (we don't have a symmetry of coefficients), abstract algebra (no group, ring, field, etc... here). I have taken the liberty to suppress them. I have added the "roots" tags.
$endgroup$
– Jean Marie
Mar 23 at 16:11